Let's first find all the way to use pennies. The only ways to use pennies at
all is to use 5 pennies exactly. So we have to make the remaining 70 cents with
4 or fewer coins. If we use no half dollar, then the only way to make the
remaining 70 cents is 2 quarters and 2 dimes.
That's 1 way.
If we use a half dollar, to make the remaining 15 cents, we can either use 1 or
2 dimes and the rest, if any, in nickels.
That's 2 ways
That exhausts all the ways to use pennies. The rest of the ways will use no
pennies, just half dollars, quarters, dimes, and nickels.
If we use no half dollar or quarters, we can use 6 or 7 dimes and the rest in
nickels.
That's 2 ways.
If we use no half dollar and 1 quarter, we can make the remaining 50 cents
using 2,3,4 or 5 dimes and the rest, if any, in nickels.
That's 4 ways.
If we use no half dollars and 2 quarters, we can make the remaining 25 cents
using 0,1, or 2 dimes and the rest, if any, in nickels.
That's 3 ways.
If we use no half dollars and 3 quarters, that's 75 cents.
That's 1 way.
If we use 1 half dollar, and no quarter, the remaining 25 cents in 3 or
fewer coins. So the only way is 0,1, or 2 dimes, with the rest in nickels.
That's 3 ways.
The only remaining way is 1 half dollar and 1 quarter. That's 1 way.
Adding all these up:
1+2+2+4+3+1+3+1 = 17 ways.
Edwin
