Question

by the end of this task, you should understand how the significance of order establishes differences between counting permutations and counting combinations.

1. herman has five different colored shirts: red, blue, yellow, purple, and gray. he will wear a different shirt each day this week from monday to friday. how many different ways can herman do this?

Explanation:

Step1: Identify the problem type

This is a permutation problem because the order in which Herman wears the shirts (different days) matters. We need to find the number of permutations of 5 shirts taken 5 at a time.

Step2: Recall the permutation formula

The formula for permutations of \( n \) objects taken \( r \) at a time is \( P(n, r)=\frac{n!}{(n - r)!} \). Here, \( n = 5 \) (number of shirts) and \( r = 5 \) (number of days, so we are arranging all 5 shirts).

Step3: Substitute the values into the formula

For \( n = 5 \) and \( r = 5 \), we have \( P(5, 5)=\frac{5!}{(5 - 5)!} \). Since \( 0!=1 \), this simplifies to \( P(5, 5)=\frac{5!}{0!}=5! \).

Step4: Calculate \( 5! \)

\( 5!=5\times4\times3\times2\times1 = 120 \).

Answer:

120