Answer: There are 32,760 color arrangements possible for painting the bedroom, garage, office, and bathroom with different colors chosen from 15 options.
Step-by-step explanation: To determine the number of color arrangements possible for painting the bedroom, garage, office, and bathroom with different colors chosen from 15 options, we need to calculate the number of permutations.
Since each room is assigned a different color, we can arrange the colors in a specific order. The number of ways to arrange 15 colors in 4 distinct positions is given by the permutation formula:
P(n, r) = n! / (n - r)!
where n is the total number of options (15 colors) and r is the number of positions (4 rooms).
Using the formula, the number of color arrangements possible is:
P(15, 4) = 15! / (15 - 4)!
= 15! / 11!
= (15 * 14 * 13 * 12) / (4 * 3 * 2 * 1)
= 32,760