# Combinations

• Dec 4th 2009, 07:09 PM
john-1
Combinations
A cookie jar contains three chocolate-chip, two peanut-butter, one lemon, one almond, and five raisin cookies.?

In how many ways can you select some cookies, if you must include at least one chocolate-chip cookie?

the answer is 216, but i don't understand why it's 216 and not 215.. because i keep getting 215.

Here's my reasoning:
It's 215 because
3 x 3 x 2 x2 6=216 BUT.. You CANNOT choose ZERO cookies, since you have to choose at least one chocolate chip cookie. Hence, choosing zero is not an option.. Now.. Subtract 1 from 216 = 215.

What's wrong with my reasoning?
• Dec 4th 2009, 10:51 PM
qmech
Your calculation seems right. You don't have to subtract 1 because the "zero cookie" case actually contains 1 chocolate chip cookie.
• Dec 5th 2009, 07:38 AM
Soroban
Hello, john-1!

Quote:

A cookie jar contains 3 chocolate-chip, 2 peanut-butter, 1 lemon, 1 almond, and 5 raisin cookies.

In how many ways can you select some cookies,
if you must include at least one chocolate-chip cookie?

Here's my reasoning:
It's 215 because: 3 x 3 x 2 x 2 x 6 = 216

BUT.. You CANNOT choose ZERO cookies, since you have to choose at least one chocolate chip cookie.
Hence, choosing zero is not an option.. Now.. Subtract 1 from 216 = 215.

What's wrong with my reasoning?

There are 3 chocolate chip cookies (C).
And you said there are three choices:
. . choose 1 C
. . choose 2 Cs
. . choose 3 Cs

See?
You've already eliminated the zero-C option.