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# counting

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29
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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?

Nov 12, 2022

#1
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Start with the bread. There are 5 types of bread and we choose 2 from them. $$\dbinom{5}{2} = \dfrac{5!}{3!\cdot2!} = 10$$ different combinations. There are 7 different ingredients and the order does not matter, so we get $$7! = 5040$$ ways to distribute the ingredients. The final answer is $$10 \cdot 5040 = 50400$$

hope this helped

Nov 13, 2022