James is making a sandwich with two slices of bread chosen from rye, wheat, white, olive loaf, foccacia, and filled with any combination of ham, cheese, lettuce, tomato, onion, mayo, mustard. If his sandwich can have one or two types of bread and the order of the ingredients doesn't matter, how many different sandwiches can James make?
+118673 one or two bread from 5 and any number of fillings from 7
one type of bread with whatever filling = \(\displaystyle 5\cdot \left[\sum_{n=1}^7\;\binom{7}{n}\right]\)
two types of bread with whatever filling = \(\displaystyle \binom{5}{2}\cdot \left[\sum_{n=1}^7\;\binom{7}{n}\right]\)
Evaluate and add
+237 7 ingredients
7c1 + 7c2 + 7c3 + 7c4 + 7c5 + 7c6 + 7c7
7 + 21 + 35 + 35 + 21 + 7 + 1
= 127 combos of ingredients
1 type of bread: 5 options
5 * 127 = 635 sandwiches
2 types of bread: 5c2 = 10
10 * 127 = 1270 sandwiches
635 + 1270 = 1905 sandwiches
