Question

jessie’s apartment has 3 rooms. she will paint each room a different color. she is choosing between 7 different paint colors. in how many different ways can jessie paint her rooms? enter the answer in the box.

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to choose and arrange 3 different colors out of 7 for 3 rooms. The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 7 \) (total number of colors) and \( r=3 \) (number of rooms to paint).

Step2: Calculate the factorial values

First, calculate \( n! = 7! = 7\times6\times5\times4\times3\times2\times1=5040 \) and \( (n - r)!=(7 - 3)!=4!=4\times3\times2\times1 = 24 \).

Step3: Apply the permutation formula

Substitute the values into the formula: \( P(7, 3)=\frac{7!}{(7 - 3)!}=\frac{5040}{24} \).

Step4: Simplify the fraction

\( \frac{5040}{24}=210 \).

Answer:

210