Final answer:
There are 20,216 ways to select 15 cookies from 6 varieties. There are 25,201 ways to select 15 cookies if at least three must be chocolate chip. There are 1,846 ways to select 15 cookies if at most 2 can be sugar cookies. There are 16,524 ways to select 15 cookies if at most 2 can be sugar cookies and at least 3 must be chocolate chip.
Step-by-step explanation:
(a) In order to find the number of ways to select 15 cookies from 6 varieties, we can use the concept of combinations. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be selected. In this case, n = 6 and r = 15. Plugging in the values, we get:
nCr = 6! / (15!(6-15)!) = 6! / (15! * (-9)!) = 6! / (15! * 9!) = (6 * 5 * 4 * 3 * 2 * 1) / ((15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) * (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)) = 20,216
(b) To find the number of ways to select 15 cookies with at least three chocolate chip cookies, we need to consider the different cases:
1. Selecting exactly 3 chocolate chip cookies: 6C3 * 3C12 = 20,160
2. Selecting exactly 4 chocolate chip cookies: 6C4 * 3C11 = 4,680
3. Selecting exactly 5 chocolate chip cookies: 6C5 * 3C10 = 360
4. Selecting exactly 6 chocolate chip cookies: 6C6 * 3C9 = 1
So, the total number of ways to select 15 cookies with at least three chocolate chip cookies is 20,160 + 4,680 + 360 + 1 = 25,201.
(c) To find the number of ways to select 15 cookies with at most 2 sugar cookies, we need to consider the different cases:
1. Selecting 0 sugar cookies: 6C0 * 3C15 = 1
2. Selecting 1 sugar cookie: 6C1 * 3C14 = 270
3. Selecting 2 sugar cookies: 6C2 * 3C13 = 1,575
So, the total number of ways to select 15 cookies with at most 2 sugar cookies is 1 + 270 + 1,575 = 1,846.
(d) To find the number of ways to select 15 cookies with at most 2 sugar cookies and at least 3 chocolate chip cookies, we can subtract the number of cases where there are more than 2 sugar cookies or less than 3 chocolate chip cookies from the total number of ways to select 15 cookies:
Total number of ways to select 15 cookies: 20,216
Number of ways to select 15 cookies with more than 2 sugar cookies: 1 + 270 + 1,575 = 1,846
Number of ways to select 15 cookies with less than 3 chocolate chip cookies: 6C0 * 3C15 + 6C1 * 3C14 + 6C2 * 3C13 = 1 + 270 + 1,575 = 1,846
Number of ways to select 15 cookies with at most 2 sugar cookies and at least 3 chocolate chip cookies: 20,216 - (1,846 + 1,846) = 16,524