There are six choices for bread, 4 choices for meat, 4 choicesfor cheese, and 12

Question

There are six choices for bread, 4 choices for meat, 4 choicesfor cheese, and 12 different garnishes. How many different sandwichpossiblities are there if you choose: a) one bread, one meat, and one cheese? ----> I think i gotthis b) one bread, one meat, one cheese, and from 0 -12garnishes? ------> I think I am stuck on thefrom 0-12...unless I was just lucky to get the rightanswer on a.    c) one bread, 0,1,2 meats; 0,1,2 cheeses; and from 0 - 12garnishes? Answers (in back): a) 96 b) 393,216 c) 2,973,696 Need to know b & c....Thank you for your time There are six choices for bread, 4 choices for meat, 4 choicesfor cheese, and 12 different garnishes. How many different sandwichpossiblities are there if you choose: a) one bread, one meat, and one cheese? ----> I think i gotthis b) one bread, one meat, one cheese, and from 0 -12garnishes? ------> I think I am stuck on thefrom 0-12...unless I was just lucky to get the rightanswer on a.    c) one bread, 0,1,2 meats; 0,1,2 cheeses; and from 0 - 12garnishes? Answers (in back): a) 96 b) 393,216 c) 2,973,696 Need to know b & c....Thank you for your time

Explanation / Answer

A) 6 * 4 * 4 = 96 B) 6 * 4 * 4 * 4096 = 393216 Note: 4096 = sum(C(12, x), x = 0..12) C) 6 * (1 + 4 + C(4, 2)) * (1 + 4 + C(4, 2)) * 4096 = 2973696 C(n, x) is used to compute the number of ways of selecting xgarnishes from the 12 options, 2 meats from the 4 options, and 2cheese from the 4 options. As a matter of fact, everything inthis problem could reduce to it since C(4, 1) = 4 (as seen whenselecting meats/cheeses), C(4, 0) = 1 (also seen when selectingmeats/cheeses), and C(6, 1) = 6 (when choosing the bread).

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