Answer: (a) If the order of the choices is not taken into consideration, we want to count the number of combinations of 6 colors chosen from 10 distinct colors. The formula for the number of combinations of k objects chosen from a set of n distinct objects is:
n choose k = n! / (k! * (n-k)!)
So, the number of ways to choose 6 colors from 10 distinct colors, without replacement and without considering the order, is:
10 choose 6 = 10! / (6! * (10-6)!) = 210
Therefore, there are 210 ways to choose 6 colors from 10 distinct colors without considering the order.
(b) If the order of the choices is taken into consideration, then each choice can be made in 10 ways, and there are 6 choices to be made. Thus, the total number of ways to choose 6 colors from 10 distinct colors, with replacement and considering the order, is:
10 * 10 * 10 * 10 * 10 * 10 = 10^6 = 1,000,000
Therefore, there are 1,000,000 ways to choose 6 colors from 10 distinct colors, considering the order.
Explanation: