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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? 

 Jul 13, 2022
 #1
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105?

I don't know what's the answer

 Jul 21, 2022
 #2
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There are \({5 \choose 1} + {5 \choose 2} = 15\) ways to choose 1 or 2 slices of bread. 

 

There are \(2^7 - 1 = 127\) ways to choose the ingredients (think of it as a binary number 0000000 to 1111111, where each digit represents an ingredient, 0 means it's not there, and 1 means it's there)

 

So, there are \(15 \times 127 = \text{___}\) sandwichs. 

 Jul 21, 2022

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