# Thread: Combinations using Binomial Theorem

1. ## Combinations using Binomial Theorem

Hello,

The problem i am having is as follows:

A deli has 6 choices for bread, 4 choices for cheese, 4 choices for meat, and 12 condiments.

How many combinations are there if you choose:
a)one bread, one meat, and one cheese?
b)one bread, one meat, one cheese, 0-12 condiments?
c)one bread, 0-2 meats, 0-2 cheeses, and 0-12 condiments?

I know that part a is 6*4*4 = 96 choices.
I also know I need to use the binomial theorem to answer pars b and c but I am not exactly sure how to use it correctly.

Thank you!

2. IF I understand the 0-12....

$\displaystyle (6)(4)(4)\sum_{x=0}^{12}{12\choose x}$

$\displaystyle =(6)(4)(4)\sum_{x=0}^{12}{12\choose x}(1)^{x}(1)^{12-x}$

$\displaystyle =(6)(4)(4)\left(1+1\right)^{12}$

$\displaystyle =(6)(4)(4)\left(2\right)^{12}$

3. then for (c)

$\displaystyle (6) \sum_{x=0}^2{4\choose x} \sum_{x=0}^2{4\choose x} \biggl(1+1\biggr)^{12}$

$\displaystyle =(6) (1+4+6)(1+4+6) (1+1)^{12}$

4. Ok, that makes sense. Thank you!