There are 210 color arrangements possible
The number of rooms (r) in the house is given as:
[tex]r = 3[/tex] ---- i.e. the kitchen, dining room, and living room
The number of available colors (n) to select from is also given as:
[tex]n = 7[/tex]
The possible color arrangement is calculated using the following permutation formula
[tex]^nP_r = \frac{n!}{(n - r)!}[/tex]
So, we have:
[tex]^7P_3 = \frac{7!}{(7 - 3)!}[/tex]
Remove bracket
[tex]^7P_3 = \frac{7!}{4!}[/tex]
Expand the numerator
[tex]^7P_3 = \frac{7 \times 6 \times 5 \times 4!}{4!}[/tex]
[tex]^7P_3 = 7 \times 6 \times 5[/tex]
[tex]^7P_3 = 210[/tex]
Hence, there are 210 color arrangements possible
Read more about permutation at:
https://brainly.com/question/11871015
