Maths Olympiad Prep

Track / Stage 4 / 56 of 340 #316 of 1964

Problem 316

AMC 12 late, AIME early
Combinatorics Difficulty 4.7 Find the answer

Example 3. In how many ways can three people be selected for three identical positions from ten candidates?

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Solution. According to formula (1.3.4), we find

C103=10!3!⋅(10−3)!=10!3!⋅7!=10⋅9⋅81⋅2⋅3=120 C_{10}^{3}=\frac{10!}{3!\cdot(10-3)!}=\frac{10!}{3!\cdot 7!}=\frac{10 \cdot 9 \cdot 8}{1 \cdot 2 \cdot 3}=120

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.