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Hello Randy,

I encountered a number sense quetion

Q: A bank contains pennies, nickels, dimes, quarters and half-dollars. How many different sets of three coins can be formed?______Answer:35

Is there any shortcut to solve this mentally?

I can explain how to get the answer 35 but unfortunately I don't have a quick mental ("in your head") way to find it.

Perhaps you already have found a way to solve it but here is the way I did it for the record.

There are 5 different types coins available in the bank. How many different sets of 3 can be made? There are 3 cases to consider:

1. All the coins in a set are of the same type. There are 5 ways that this can happen (each one of the 5 coins is used to form the 3-coin set). So this contributes 5 sets to the total.

2. Two coins are the same in a set. For each of the 5 2-coins-the-same sets, the 3rd member of the set can be chosen from the remaining 4 types of coins. Thus, each 2-coin set has 4 ways of being chosen, which gives 4 times the 5 2-coin set or 20 ways.

3. Only 1 type of coin is allowed in a set. This is then the classic combination problem of 5 objects taken 3 at a time given by the binomial formula 5!/( 3!2!) = 5･4･3･2/(3･2･2) = 10.

Adding up the numbers for these (mututally exclusive) cases gives 5 + 20 + 10 = 30.

Hope this helps.

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