1. ## sandwich combinations

Choices

4 meat
4 cheese
12 garnishes

How many different possibilities for sandwiches are there is you choose
a- 1 bread, 1 meat, 1 cheese, and no garnishes?
b- 1 bread, 1 meat, 2 DIFFERENT cheeses, and from 0 to 12 garnishes?

Thanks

2. multiplication rule
(a) 6 times 4 times 4
(b) 6 times 4 times 6 because (4 choose 2 =6 cheeses) the garnishes, well thats huge
the number of garnishes is the sum of 12 choose x from 0 to 12

$\sum_{x=0}^{12}{12 \choose x}= \sum_{x=0}^{12}{12 \choose x}1^x1^{12-x}= 2^{12}$
via the binomial theorem

3. Thank you.

part a=96
partb=6*4*6*4096=589824?

4. Yes, if I understood this correctly
and by the way, who cut the cheese?