SOLUTION: How many ways can I make $1 using 5cent, 20cent and 50cent coins

Question 1141278: How many ways can I make $1 using 5cent, 20cent and 50cent coins
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

I'll let you find the actual answer; it should be relatively easy, and the number is small.

Let me demonstrate the strategy for solving the problem using different numbers.

My example: find the number of ways of making $1 using quarters (25 cents), dimes (10 cents) and pennies (1 cent).

The strategy:

(1) determine the possible numbers of quarters.

4 quarters makes a dollar, so the maximum number of quarters is 4. The possible numbers of quarters are 4, 3, 2, 1, and 0.

(2) For each of the possible numbers of quarters, determine the possible numbers of dimes. You don't need to worry about the pennies; whatever is left over after the quarters and dimes cam be made by the pennies.

(a) 4 quarters. That makes the whole dollar. There is only one choice for the number of dimes: 0. Number of solutions using 4 quarters: 1.

(b) 3 quarters. That makes 75 cents; there is 25 cents left. The number of dimes can be 0, 1, or 2. Number of solutions using 3 quarters: 3.

(c) 2 quarters. That makes 50 cents; there is 50 cents left. The number of dimes can be 0, 1, 2, 3, 4, or 5. Number of solutions using 2 quarters: 6.

(d) 1 quarter. That is 25 cents; there is 75 cents left. The number of dimes can be any number from 0 to 7. Number of solutions using 1 quarter: 8.

(e) 0 quarters. The number of dimes can be any number from 0 to 10. Number of solutions using 0 quarters: 11.

Total number of ways of making $1 using 25 cent, 10 cent, and 1 cent coins: 1+3+6+8+11 = 29.

Use the same strategy for your problem using 50 cent, 20 cent, and 5 cents coins.

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In response to a query from the reader....

In your problem, the 5 cent coins are like the pennies in my similar example -- you don't need to worry about them, because anything left after the 50 cent and 20 cent coins can be made up by the 5 cent coins.

So to find the answer to your problem you only need to look at the different possible numbers of 50 cent coins, and then find the number of choices for the number of 20 cent coins that go with each possible number of 50 cent coins.
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