Question

there are 25 people competing in a rubber duck regatta. each person is allowed only one entry. if there are no ties, how many ways are there to have the first, second, and third place winners?
a 14,380 ways
b 13,800 ways
c 75 ways
d 15,625 ways

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 3 winners (first, second, third) out of 25 people. The formula for permutations of \( n \) objects taken \( r \) at a time is \( P(n, r)=\frac{n!}{(n - r)!} \). Here, \( n = 25 \) and \( r = 3 \).

Step2: Apply the permutation formula

First, calculate \( n! = 25! \) and \( (n - r)!=(25 - 3)!=22! \). Then, \( P(25, 3)=\frac{25!}{22!} \). Since \( 25! = 25\times24\times23\times22! \), we can cancel out \( 22! \) from the numerator and denominator. So, \( P(25, 3)=25\times24\times23 \).

Step3: Calculate the product

\( 25\times24 = 600 \), and then \( 600\times23 = 13800 \).

Answer:

B. 13,800 ways