Question

six teammates are competing for first, second, and third place in a race. how many possibilities are there for the top three positions? 30 20 120 240

Explanation:

Step1: Identify the problem type

This is a permutation problem since the order of the teammates (first, second, third place) matters. The formula for permutations of \( n \) items taken \( r \) at a time is \( P(n, r)=\frac{n!}{(n - r)!} \). Here, \( n = 6 \) (total teammates) and \( r = 3 \) (top three positions).

Step2: Calculate the permutation

First, find \( n! \) and \( (n - r)! \). \( n!=6!=6\times5\times4\times3\times2\times1 = 720 \), and \( (n - r)!=(6 - 3)!=3!=3\times2\times1 = 6 \). Then, \( P(6, 3)=\frac{6!}{(6 - 3)!}=\frac{720}{6}=120 \).

Answer:

120 (corresponding to the option with 120)