Question

1. if john has pairs of red, orange, yellow, blue, and green socks, how many orders can he wear them in over five days? repetition is allowed because john is alright with wearing dirty socks.

25
5!
5⁵
125

Explanation:

Step1: Identify the problem type

This is a permutation with repetition problem. We need to find the number of ways to choose a sock color for each of the 5 days, where repetition is allowed.

Step2: Determine the number of choices per day

John has 5 colors of socks (red, orange, yellow, blue, green). So, for each day, he has 5 choices.

Step3: Calculate the total number of orders

Since there are 5 days and 5 choices each day, and repetition is allowed, we use the formula for permutations with repetition: \( n^r \), where \( n \) is the number of choices and \( r \) is the number of trials (days). Here, \( n = 5 \) and \( r = 5 \), so the total number of orders is \( 5^5 \).

Answer:

\( 5^5 \) (Option: \( 5^5 \))