FREE AIME Combinatorics Questions and Answers

Question 1

How many ways can 5 people be seated in a row?

Accepted Answer

120

Answer

The number of ways to arrange 𝑛 people in a row is 𝑛! n!. For 5 people, this is 5!=5×4×3×2×1=120.

Question 2

In how many ways can a committee of 3 people be chosen from a group of 10?

Accepted Answer

120

Answer

The number of ways to choose 𝑘 people from 𝑛 is given by 120

Question 3

How many ways can the letters of the word "AIME" be arranged?

Accepted Answer

24

Answer

The number of ways to arrange the letters of a word with all distinct letters is 𝑛! For "AIME", which has 4 letters, the number of arrangements is 4!=4×3×2×1=24.

Question 4

How many ways can you distribute 5 identical candies to 3 children so that each child gets at least one candy?

Accepted Answer

10

Answer

This is a problem of distributing indistinguishable objects (candies) into distinguishable bins (children) with the restriction that each bin gets at least one object.
This is solved using the stars and bars method. We first give each child one candy, then distribute the remaining 2 candies among the 3 children. The number of ways is given by 6.

Question 5

In how many ways can you choose 2 different cards from a standard deck of 52 cards?

Accepted Answer

1326

Answer

The number of ways to choose 2 cards from 52 is given by 1326

AIME - American Invitational Mathematics Examination Practice Test

AIME - American Invitational Mathematics Examination Practice Test

FREE AIME Combinatorics Questions and Answers