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The university philosophy department is holding a dinner for a select group of professors and students, each of whom has exactly one seat at exactly one of the three available tables-- 1, 2, and 3. There are four professors-- W, X, Y, and Z-- and five students-- B, C, D, E, and F. The seating arrangements must adhere to the following conditions without exception:
There is at least one professor and one student at each of the three tables.
Two of the tables seat two individuals, and one of the tables seats five individuals.
If W is at a table with C, then E is at a table with B.
If D is at a table with B, then X is not sitting with E.
F is never sitting at a table with more than one professor.
W always sits at the table with the most seated individuals.
Y is always sitting at a higher-numbered table than Z.
Which of the following is an acceptable seating arrangement?
This is a relatively easy question that simply tests your basic grasp of the game rules. Each of the incorrect answers breaks one or more rules in some way.
Table 1: Y, W, C, E, B
Table 2: Z, D
Table 3: X, F
This arrangement breaks the rule that Y must be at a higher numbered table than Z.
Table 1: X, W, C, D
Table 2: Z, E, B
Table 3: Y, F
This arrangement breaks the rule that there must always be more students than professors at any given table. There are an equal number of professors and students at Table 1.
Table 1: W, Z, C, D, F
Table 2: Y, E
Table 3: X, B
This arrangement breaks the rule that F never sits with more than one professor.
Table 1: W, X, C, B, D
Table 2: Z, E
Table 3: Y, F
This arrangement breaks the rule that E and B sit at the same table if W and C do.
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The university philosophy department is holding a dinner for a select group of professors and students, each of whom has exactly one seat at exactly one of the three available tables-- 1, 2, and 3. There are four professors-- W, X, Y, and Z-- and five students-- B, C, D, E, and F. The seating arrangements must adhere to the following conditions without exception:
There is at least one professor and one student at each of the three tables.
Two of the tables seat two individuals, and one of the tables seats five individuals.
If W is at a table with C, then E is at a table with B.
If D is at a table with B, then X is not sitting with E.
F is never sitting at a table with more than one professor.
W always sits at the table with the most seated individuals.
Y is always sitting at a higher-numbered table than Z.
If W is seated at Table 3, then which two individuals MUST be seated at the same table?
It is possible to simply brute force this answer, however, it is possible to reason it out as well.
W is at the largest table, and the largest table always has two professors and three students (since the large table seats 5 and must accommodate two professors in order to get all of the professors seated with at least one professor and one student at each table). Z may never sit at Table 3, since Y must be seated at a higher table.
F never sits with more than one professor, so he doesn't sit at this table (and he also never sits with W). Thus, W sits with three of B, C, D, and E. If W sits with C, then E and B must also sit together. Groups of multiple students must sit with W. So, Table 3 MUST contain C, E and B if C is seated with W. Either X or Z may sit as well, but that doesn't really matter for this question.
If W does NOT sit with C, then we are left with D, E, and B which can also make valid diagrams.
Between C, E, and B and D, E, and B, E and B are the common factors. W, E, and B always sit together (this is, in fact, true regardless of whether or not they sit at Table 3).
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The university philosophy department is holding a dinner for a select group of professors and students, each of whom has exactly one seat at exactly one of the three available tables-- 1, 2, and 3. There are four professors-- W, X, Y, and Z-- and five students-- B, C, D, E, and F. The seating arrangements must adhere to the following conditions without exception:
There is at least one professor and one student at each of the three tables.
Two of the tables seat two individuals, and one of the tables seats five individuals.
If W is at a table with C, then E is at a table with B.
If D is at a table with B, then X is not sitting with E.
F is never sitting at a table with more than one professor.
W always sits at the table with the most seated individuals.
Y is always sitting at a higher-numbered table than Z.
If W and X are sitting together, then how many possible valid seating arrangements are there?
If W and X are sitting together, then they must also be sitting with C, E, and B. (See previous answer explanation.)
Y always sits at a higher table than Z, so no matter which table W and X are sitting at, there is only one possible arrangement for Y and Z. (e.g. If W and X sit at Table 1, then Z sits at Table 2 and Y sits at Table 3. If W and X sit at Table 2, then Z sits at Table 1 and Y sits at Table 3, etc.)
The remaining letters, D and F can both sit with either Y or Z. Thus, there are two possible valid seating arrangements if X and W sit at the same table.
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The university philosophy department is holding a dinner for a select group of professors and students, each of whom has exactly one seat at exactly one of the three available tables-- 1, 2, and 3. There are four professors-- W, X, Y, and Z-- and five students-- B, C, D, E, and F. The seating arrangements must adhere to the following conditions without exception:
There is at least one professor and one student at each of the three tables.
Two of the tables seat two individuals, and one of the tables seats five individuals.
If W is at a table with C, then E is at a table with B.
If D is at a table with B, then X is not sitting with E.
F is never sitting at a table with more than one professor.
W always sits at the table with the most seated individuals.
Y is always sitting at a higher numbered table than Z.
Which of the following two individuals can never sit at the same table together?
Because F never sits at the big table, W must sit with one other professor and three students out of B, C, D, and E. If C and D were to sit together, they MUST sit at the large table with W, since every other table seats one professor and one student. If C sits with W, then E and B must sit with W as well, since, again, no other table can accommodate a block of multiple students. Since there must be two professors at the large table, W, ?, C, B, and E completely fills the table with no room for D. C and D can never sit together.
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The university philosophy department is holding a dinner for a select group of professors and students, each of whom has exactly one seat at exactly one of the three available tables-- 1, 2, and 3. There are four professors-- W, X, Y, and Z-- and five students-- B, C, D, E, and F. The seating arrangements must adhere to the following conditions without exception:
There is at least one professor and one student at each of the three tables.
Two of the tables seat two individuals, and one of the tables seats five individuals.
If W is at a table with C, then E is at a table with B.
If D is at a table with B, then X is not sitting with E.
F is never sitting at a table with more than one professor.
W always sits at the table with the most seated individuals.
Y is always sitting at a higher numbered table than Z.
Which of the following conditions, if added to the existing set of conditions, would still allow for a valid seating arrangement?
If W and D must always sit together, that simply collapses the possibilities for the large table to only W, Y, E, B, D. This is a valid seating arrangements.
\[There are never more than two students seated together at a table.\] Three students MUST be seated at the large table in order to accommodate all of the professors while maintaining the "at least one professor and one student at each table" rule.
\[E and F must be seated together.\] F always sits with one professor. E is not a professor, so F and E is not a valid seating arrangement for a small table with two seats. The large table with five seats MUST sit two professors in order to accomodate all of the professors.
\[B is never seated at a table with more than one professor.\] E and B MUST sit at the large table with W. The large table must seat two professors in order to accomodate them all.
\[If W and X are seated together at the same table, then C is seated with them at that table as well.\] If W and X are seated at the same table, then that table MUST have the arrangement W, X, C, E, B. If the arrangement was W, X, D, E, B, then we would be breaking the rule that X and E cannot sit together if D and B do.
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A museum must choose five stuffed birds to be shown in their newest exhibit. Their choices include an Albatross, a Bluebird, a Condor, a Diver, an Eagle, a Flamingo, a Gull and a Hummingbird. They must choose the group according to the following restrictions:
If the Diver is chosen, the Albatross is not
The Eagle is chosen only if the Flamingo is not
If the Flamingo is chosen, the Albatross and the Hummingbird are also chosen
If the Gull is chosen, either the Hummingbird or the Bluebird are chosen, but not both
Which of the following is a complete and accurate list of birds that could be shown in the exhibit?
From the first two rules we know that the Diver and the Albatross, and the Eagle and the Flamingo can never appear in the same group together, which eliminates two choices immediately. We also know that if the Flamingo is chosen, both the Albatross and the Hummingbird must also be chosen, eliminating another incorrect answer. The last rule tells us that if the Gull is chosen, either the Hummingbird OR the Bluebird must be chosen, but not both. Therefore, we can deduce that if both the Hummingbird AND Bluebird are chosen, the Gull is not. This eliminates the last contender, leaving only the correct answer.
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A museum must choose five stuffed birds to be shown in their newest exhibit. Their choices include an Albatross, a Bluebird, a Condor, a Diver, an Eagle, a Flamingo, a Gull and a Hummingbird. They must choose the group according to the following restrictions:
If the Diver is chosen, the Albatross is not
The Eagle is chosen only if the Flamingo is not
If the Flamingo is chosen, the Albatross and the Hummingbird are also chosen
If the Gull is chosen, either the Hummingbird or the Bluebird are chosen, but not both
If the Flamingo is chosen, how many different groups are possible?
If the Flamingo is chosen, we immediately also choose the Albatross and the Hummingbird. This leaves only two spots to be filled. We can eliminate the Eagle because it cannot appear with the Flamingo, and we can eliminate the Diver because it cannot appear with the Albatross. We are left to choose from the Bluebird, the Condor and the Gull. If we choose the Gull, we cannot also choose the Bluebird - since the Hummingbird is already present, this situation would violate the rule involving the Gull. Therefore, we must chose either the Gull OR the Bluebird, and then flll the remaining spot with the Condor. The two distinct possibilities are: Flamingo, Albatross, Hummingbird, Condor and Gull OR Flamingo, Albatross, Hummingbird, Condor and Bluebird.
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A museum must choose five stuffed birds to be shown in their newest exhibit. Their choices include an Albatross, a Bluebird, a Condor, a Diver, an Eagle, a Flamingo, a Gull and a Hummingbird. They must choose the group according to the following restrictions:
If the Diver is chosen, the Albatross is not
The Eagle is chosen only if the Flamingo is not
If the Flamingo is chosen, the Albatross and the Hummingbird are also chosen
If the Gull is chosen, either the Hummingbird or the Bluebird are chosen, but not both
If the Diver is chosen, which of the following must be true?
Knowing that the Diver is chosen allows us to immediately eliminate the Albatross according to the first rule. Eliminating the Albatross causes us to also eliminate the Flamingo in accordance with the second rule (the contrapositive of this rule states that if either the Albatross or the Hummingbird is NOT chosen, the Flamingo is also not chosen). We know we can only choose two out of the Gull, the Hummingbird and the Bluebird because of the last rule. This leaves two final empty spots which must be filled by the Eagle and the Condor.
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A museum must choose five stuffed birds to be shown in their newest exhibit. Their choices include an Albatross, a Bluebird, a Condor, a Diver, an Eagle, a Flamingo, a Gull and a Hummingbird. They must choose the group according to the following restrictions:
If the Diver is chosen, the Albatross is not
The Eagle is chosen only if the Flamingo is not
If the Flamingo is chosen, the Albatross and the Hummingbird are also chosen
If the Gull is chosen, either the Hummingbird or the Bluebird are chosen, but not both
If the Hummingbird is NOT chosen all of the following must be true EXCEPT:
When we eliminate the Hummingbird we must also immediately eliminate the Flamingo. This leaves us with six leftover variables to choose from. Since the Diver and the Albatross can never be chosen together, we know that the four other variables, namely the Gull, the Blurbird, the Condor and the Eagle all must be chosen. The only option is whether to choose the Albatross or the Diver to complete the group.
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A class consists of eight students, each with either black, brown, red, or blonde hair. The following statements accurately characterize the students in the class:
1. Four girls have either blonde or brown hair
2. Only one student has red hair
3. Two boys have the same hair color
4. There are exactly three students in the class with brown hair
5. There is at least one type of each hair color represented in the class
What can we say about the distribution of students with black hair?
We are trying to figure out if it is possible for four or more students to have black hair, since we know there are exactly three students with brown hair. Since three of the eight students have brown hair, that leaves five. Based on the second and fifth condition we know one of those five has red hair, which leaves four. Based on the fifth conditon we know there must be at least one blonde, which leaves three. Therefore, the absolute maximum number of students with black hair would have to be three, which is not more than the number of students with brown hair.
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A class consists of eight students, each with either black, brown, red, or blonde hair. The following statements accurately characterize the students in the class:
1. Four girls have either blonde or brown hair
2. Only one student has red hair
3. Two boys have the same hair color
4. There are exactly three students in the class with brown hair
5. There is at least one type of each hair color represented in the class
What is the maximum number of students with blonde hair in the class?
We know that of the eight students, three of those students have brown hair, which leaves five. Based on the second and fifth conditions, we know that one of those students has red hair, which leaves four. Based on the fifth condition, we know that at least one of those students has black hair, which leaves three. Therefore, the absolute maximum number of students with blonde hair is three.
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A class consists of eight students, each with either black, brown, red, or blonde hair. The following statements accurately characterize the students in the class:
1. Four girls have either blonde or brown hair
2. Only one student has red hair
3. Two boys have the same hair color
4. There are exactly three students in the class with brown hair
5. There is at least one type of each hair color represented in the class
What can we NOT say for certain about the distribution of girls in the class?
There is at least one possible scenario in which no girl has the same colored hair as any of the boys in the class. It is possible, for instance, that there are only two boys in the classroom. In this case, based on the third condition we know that the boys will have the same color hair. If the two boys have black hair, then that leaves six girls. Based on the first condition, we know that four girls have either blonde or brown hair, neither of which is black. That leaves two girls. Based on the second and fifth condition we know that one of those girls has red hair, which leaves one girl. It is not specified that ONLY four girls have brown or blonde hair, which means that more than four girls could have either blonde or brown hair. If the final girl has blonde or brown hair, then no girl has black hair, and therefore no girl has the same colored hair as any of the boys in the classroom.
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A class consists of eight students, each with either black, brown, red, or blonde hair. The following statements accurately characterize the students in the class:
1. Four girls have either blonde or brown hair
2. Only one student has red hair
3. Two boys have the same hair color
4. There are exactly three students in the class with brown hair
5. There is at least one type of each hair color represented in the class
If there are three boys in the class, which of the following MUST be true?
Based on the first condition, we know that four girls have either blonde or brown hair. Based on the second condition and the condition posed within this answer choice, we are assuming that the one student with red hair is a girl. This means that there are three students left. Based on the condition posed within this question, we know that those three students must be boys. Based on the fifth premise, we know that at least one of each hair color must be represented. Since black hair is not represented by any of the girl students, we know that at least one of the boys must have black hair.
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A class consists of eight students, each with either black, brown, red, or blonde hair. The following statements accurately characterize the students in the class:
1. Four girls have either blonde or brown hair
2. Only one student has red hair
3. Two boys have the same hair color
4. There are exactly three students in the class with brown hair
5. There is at least one type of each hair color represented in the class
Which of the following is a possible scenario for the class distribution?
This is the only scenario that meets all of the conditions: four girls have either blonde or brown hair (three have blonde hair and one has brown), only one student has red hair, two boys have the same hair color (brown), there are exactly three students with brown hair (two boys and one girl), and at least one of every hair color is represented.
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A class consists of eight students, each with either black, brown, red, or blonde hair. The following statements accurately characterize the students in the class:
1. Four girls have either blonde or brown hair
2. Only one student has red hair
3. Two boys have the same hair color
4. There are exactly three students in the class with brown hair
5. There is at least one type of each hair color represented in the class
If two boys have blonde hair, which of the following MUST be true?
Based on the fourth premise, we know that exactly three students have brown hair. If from the first premise we know that there are at least four girls, then that means there can be a maximum of four boys. If two of those boys have blonde hair, and based on the fourth premise we know that there are three students with brown hair, then even if the other two boys in the class has brown hair, at least one girl would have to have brown hair as well.
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A successful recruiting drive by the local laser tag league has lead to the recruitment of seven new recruits-- P, Q, R, S, T, U, and V. After completing their initial training, they will all be assigned to one of the laser tag league's four teams-- Dragon, Salamander, Phoenix, and Rabbit. Their assignment to the teams must conform to the following conditions:
At least one new member is assigned to each of the four teams.
Q and T are not assigned to the same team if R and V are not assigned to the same team.
If T is not assigned to Dragon, then U is not assigned to Dragon.
P is always on a team with more new recruits than S.
Dragon gets more new recruits than any other team.
Which of the following is an acceptable assignment of new recruits to the four teams?
Dragon: Q, T, V
Salamander: R
Phoenix: S
Rabbit: P, U
This one breaks the rule that QT cannot be paired together if RV are not paired together. This is more clear if you write out the contrapositive.
~(RV) --> ~(QT)
(QT) --> (RV)
Dragon: V, P
Salamander: S
Phoenix: Q, R
Rabbit: T, U
This is incorrect, because it breaks the rule that Dragon must have more new recruits than any other team, not simply as many. This incorrect answer, if properly considered, also leads to the key insight that Dragon will always have at least three new recruits. That is because the distributions of new members, in order to satisfy both rules that Dragon must have the most and each team must have at least one, will look like either 4/1/1/1 or 3/2/1/1. There are no other possible distributions. Furthermore, if it is a 3/2/1/1 distribution and P is NOT a Dragon, then P is in the team with two new recruits, since P must be in a team with more new recruits than S (who must be a team with only one new recruit).
Dragon: R, V, Q, T
Salamander: U
Phoenix: P
Rabbit: S
This is incorrect, because S and P are on teams that have the same number of new recruits. P must be on a team with more.
Dragon: P, V, U, Q
Salamander: T
Phoenix: R
Rabbit: S
This is incorrect because T is not on Dragon, but U is on Dragon. This is another one that becomes more clear if you write out the contrapositive.
~(Td) --> ~(Ud)
Ud --> Td
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A successful recruiting drive by the local laser tag league has lead to the recruitment of seven new recruits-- P, Q, R, S, T, U, and V. After completing their initial training, they will all be assigned to one of the laser tag league's four teams-- Dragon, Salamander, Phoenix, and Rabbit. Their assignment to the teams must conform to the following conditions:
At least one new member is assigned to each of the four teams.
Q and T are not assigned to the same team if R and V are not assigned to the same team.
If T is not assigned to Dragon, then U is not assigned to Dragon.
P is always on a team with more new recruits than S.
Dragon gets more new recruits than any other team.
Which of the following cannot be true?
\[Q is assigned to Dragon and U is assigned to Dragon.\]
If U is assigned to Dragon, then T must also be assigned to Dragon. This is the contrapositive of the rule that U is not in Dragon if T is not in Dragon.
Dragon: Q, U, T
Salamander:
Phoenix:
Rabbit:
If Q and T are together, then R and V must also be together. This is the contrapositive of the rule that if R and V are not together, then Q and T are not together. Now, we can't put both R and V in Dragon, because then there wouldn't be enough recruits to distribute to other teams. For now, let's put them in Salamander.
Dragon: Q, U, T
Salamander: R, V
Phoenix:
Rabbit:
We must still deal with P and S. P must be in a group with more new recruits than S, so it it would have to go in Dragon or Salamander. Unfortunately, that would mean that either Phoenix or Rabbit would be left without a recruit, which is unacceptable.
Thus, we know that Q and U cannot both be in Dragon together.
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A successful recruiting drive by the local laser tag league has lead to the recruitment of seven new recruits-- P, Q, R, S, T, U, and V. After completing their initial training, they will all be assigned to one of the laser tag league's four teams-- Dragon, Salamander, Phoenix, and Rabbit. Their assignment to the teams must conform to the following conditions:
At least one new member is assigned to each of the four teams.
Q and T are not assigned to the same team if R and V are not assigned to the same team.
If T is not assigned to Dragon, then U is not assigned to Dragon.
P is always on a team with more new recruits than S.
Dragon gets more new recruits than any other team
Which of the following, if true, means that P must be assigned to Dragon?
\[Q is assigned to Salamander and T is assigned to Salamander.\]
If this is the case, then P must be assigned to Dragon. This mostly tests your ability to deal with the ~(RV) --> ~(QT) rule, and its contrapositive, (QT) --> (RV).
d:
s: Q, T
p:
r:
To proceed from here, you really need to have written down the contrapositive mentioned above into your rules. With that contrapositive, you know that R and V must be on a team together. Since, Dragon must have the most new recruits, R and V must both go to Dragon, and there will be exactly one other new recruit that joins them there.
d: R, V, _
s: Q, T
p: _
r: _
T is not assigned to Dragon, so U cannot be assigned to Dragon. Let's just put U in Phoenix.
d: R, V, _
s: Q, T
p: U
r: _
That leaves S and P. S cannot be assigned to Dragon, because there would be no way for P to join a team with more new recruits. Thus, S must join Rabbit. That means that P must be assigned to Dragon.
d: R, V, P
s: Q, T
p: U
r: S
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A successful recruiting drive by the local laser tag league has lead to the recruitment of seven new recruits-- P, Q, R, S, T, U, and V. After completing their initial training, they will all be assigned to one of the laser tag league's four teams-- Dragon, Salamander, Phoenix, and Rabbit. Their assignment to the teams must conform to the following conditions:
At least one new member is assigned to each of the four teams.
Q and T are not assigned to the same team if R and V are not assigned to the same team.
If T is not assigned to Dragon, then U is not assigned to Dragon.
P is always on a team with more new recruits than S.
Dragon gets more new recruits than any other team.
If Q and T are both assigned to the same team and only one new recruit is assigned to Rabbit, then which of the following is a complete list of the recruits that could possibly be assigned to Rabbit?
If Q and T are paired together, then R and V must be paired together, thanks to the contrapositive to the rule regarding those pairs. If that's the case, then we know that P is assigned to one of the two teams to which R and V and Q and T are assigned, of which one is Dragon. Since Rabbit is only getting one new recruit, then we know that P, Q, T, R, and V are not going to be available to be assigned to Rabbit. That leaves only S and U, neither of which are restricted in this case from being assigned to any particular teams, only from being assigned to the teams that are already made full by the QT and RV pairs.
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A successful recruiting drive by the local laser tag league has lead to the recruitment of seven new recruits-- P, Q, R, S, T, U, and V. After completing their initial training, they will all be assigned to one of the laser tag league's four teams-- Dragon, Salamander, Phoenix, and Rabbit. Their assignment to the teams must conform to the following conditions:
At least one new member is assigned to each of the four teams.
Q and T are not assigned to the same team if R and V are not assigned to the same team.
If T is not assigned to Dragon, then U is not assigned to Dragon.
P is always on a team with more new recruits than S.
Dragon gets more new recruits than any other team.
If P is assigned to Salamander, which of the following must be false?
If P is assigned to Salamander, then you know you're dealing with a 3/2/1/1 distribution of the recruits. Furthermore, you know that Q and T can never be placed in the same team, because there would be no place to put R and V. S must be on one of the teams that only recruits one person, meaning it is not on Dragon nor is it with P (although P and S are never together anyway).
d: _ _ _
s: P, _
p: S
r: _
This means we need to deal with Q, R, T, U, and V. We know that U cannot be in Dragon unless T is. That means we're going to have two separate diagrams-- one with T and U in Dragon and one where that is not the case.
d: R, U, T
s: P, V
p: Q
r: S
However, when we actually attempt to make a diagram without both T and U in Dragon, we run into a problem.
d: Q, R, U?
s: P, V
p: S
r: T
d: Q, R, T?
s: P, V
p: S
r: U
U cannot be in Dragon if T is not in Dragon. If U is not in Dragon, then either T or V must take up the slack. However, putting T in Dragon is problematic, then you have a QT pair which requires you to do R, V, which is impossible given the constraints.
Thus, the diagram without T and U in Dragon looks like the following:
d: Q, R, V
s: P, U
p: S
r: T
There are some slight variations in these diagrams, but, when you compare it against the answer choices, you'll see that \[Q is assigned to Dragon and V is assigned to Salamander.\] is an impossibility. In order to place those, you're forced into a situation where Q and T are on the same team, but there is no way to put R and V together, since P and V ae already together.
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