Question

example 5: in how many ways can a president, secretary and treasurer be chosen from a student council of 20 students?

Explanation:

Step1: Identify the problem type

This is a permutation problem because the order of selection (president, secretary, treasurer) matters. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 20 \) (total students) and \( r = 3 \) (positions to fill).

Step2: Apply the permutation formula

Substitute \( n = 20 \) and \( r = 3 \) into the formula:
\( P(20, 3)=\frac{20!}{(20 - 3)!}=\frac{20!}{17!} \)

Step3: Simplify the factorial expression

Since \( n!=n\times(n - 1)\times\cdots\times1 \), we can write \( 20! = 20\times19\times18\times17! \). Then,
\( \frac{20\times19\times18\times17!}{17!}=20\times19\times18 \)

Step4: Calculate the result

\( 20\times19 = 380 \), and \( 380\times18 = 6840 \)

Answer:

6840