What is the minimum amount of handshakes that can occur among fifteen people in a meeting, if each person only shakes each other person's hand once?

Claus is taking his twin brother Lucas out for ice cream. Claus knows that his brother is indecisive and wants to spend as little time choosing ice cream as possible. Claus can choose how many scoops Lucas can make for a sundae, as long as Lucas gets at least four. If there are twelve ice cream options, how many scoops should Claus tell Lucas to get?

Each scoop of ice cream is a unique flavor.

There are 20 people eligible for town council, which has three elected members.

Quantity A

The number of possible combinations of council members, presuming no differentiation among office-holders.

Quantity B

The number of possible combinations of council members, given that the council has a president, vice president, and treasurer.

Clark is in the market for new capes. If the cape store sells 48 unique types of capes, a purchase of how many capes will correspond to the minimum amount of potential combinations of capes?

Mohammed is being treated to ice cream for his birthday, and he's allowed to build a three-scoop sundae from any of the thirty-one available flavors, with the only condition being that each of these flavors be unique. He's also allowed to pick different toppings of the available , although he's already decided well in advance that one of them is going to be peanut butter cup pieces.

Knowing these details, how many sundae combinations are available?

Quantity A: The number of possible combinations when three choices are made from six options.

Quantity B: The number of possible permutations when three choices are made from four options.

There are 300 people at a networking meeting. How many different handshakes are possible among this group?

Lisa is dressing warm for the winter. She'll be layering three shirts over each other, and two pairs of socks. If she has fifteen shirts to choose from, along with ten different kinds of socks, how many ways can she layer up?

What is the number of possible 4-letter words that can be made from the 26 letters in the alphabet, where all 4 letters must be different?

Assume non-sensical words count, i.e. "dnts" would count as a 4-letter word for our purposes.

Jack is putting together his winter ensemble to take with him on a weekend ski trip. He'll be bringing two scarves out of a selection of twelve, four pairs of socks from a group of ten, and three sweaters from a choice of six. How many clothing combinations are available to him?