Brian wants to buy a muffin that costs $0.80. How many different combinations of nickels and dimes can be used to make $0.80?

Just count them:

N D 16 0 14 1 12 2 10 3 8 4 6 5 4 6 2 7 0 8

Not criticizing, just curious. Would 16 N & 0 D, and 0 N & 8 D, be considered combinations, since there is only one type of coin used?

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Yes i believe they should be included. Even though there is a zero count of one of them, each one is still a valid combination.