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 not allowed because john puts his socks in the wash at the end of the day.

$5^5$
25
125
$5!$

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 5 different items (sock pairs) over 5 days with no repetition.

Step2: Recall the permutation formula

The number of permutations of \( n \) distinct objects taken \( n \) at a time is given by \( n! \) (n factorial), where \( n! = n\times(n - 1)\times(n - 2)\times\cdots\times1 \). Here, \( n = 5 \) (since there are 5 sock pairs and 5 days).

Step3: Calculate the result

For \( n = 5 \), the number of permutations is \( 5! = 5\times4\times3\times2\times1 = 120 \), and \( 5! \) is the notation for this factorial. The other options: \( 5^5 = 3125 \), \( 25 = 5\times5 \), \( 125 = 5^3 \) do not represent permutations of 5 distinct items.

Answer:

\( 5! \) (the option with \( 5! \))