SOLUTION: How many ways can you make $0.25 using pennies, nickels, dimes, and quarters

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Question 257726: How many ways can you make $0.25 using pennies, nickels, dimes, and quarters
Found 3 solutions by drk, richwmiller, Edwin McCravy:
Answer by drk(1908)   (Show Source): You can put this solution on YOUR website!
There appear to be 13 ways. Here they are in {p,n,d,q} form:
{p,n,d,q}
{0,0,0,1}
(0,1,2,0}
{5,0,2,0}
{0,3,1,0}
{5,2,1,0}
{10,1,1,0}
{15,0,1,0}
{0,5,0,0}
{5,4,0,0}
{10,3,0,0}
{15,2,0,0}
{20,1,0,0}
{25,0,0,0}
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
one way with q 1q
2 dimes and a nickel
2 dimes and 5 pennies
1 dime 3 n
1d 2n 5p
25=
1q=1
+(2d+5) 1n or 5p 2
+(1d+15) n+10p 2n+5p 2
+ 5n 1
+(4n+5) 1
+(2n+15) d+5p 1
+25p 1
+20p+5 1
+15p+10 +d +2n 2
+10p+15 d+n 1
+(5p+20) 2d 1d+2n 2
15 if various combos of individual pennies, nickels and dimes are counted as 1
Not interested in counting if each penny is counted as a separate way.
Then the 25 pennies become 25!

So there is probably some duplication here .

Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!

p + 5n + 10d + 25q = 25

There is just one way to have 25 cents with only 1 quarter.

1.  1 quarter = $0.25

So for all the other ways we only have pennies, nickels and dimes.

p + 5n + 10d = 25

The number of pennies must be a multiple of 5 (including 0)
So let p = 5k where k is an integer 

5k + 5n + 10d = 25

Divide through by 5

k + n + 2d = 5

     k + n = 5 - 2d

So d can only be 0 thru 2 and have the right side positive.

For d = 0, we have

k + n = 5, there are 6 possibilities, k = 0 thru 5 and n = 5 thru 0 

2.  k=0, 0 pennies, 5 nickels, 0 dimes = $0.25
3.  k=1, 5 pennies, 4 nickels, 0 dimes = $0.25
4.  k=2, 10 pennies, 3 nickels, 0 dimes = $0.25
5.  k=3, 15 pennies, 2 nickels, 0 dimes = $0.25
6.  k=4, 20 pennies, 1 nickels, 0 dimes = $0.25
7.  k=5, 25 pennies, 0 nickels, 0 dimes = $0.25


For d = 1, we have

k + n = 3, there are 4 possibilities, k = 0 thru 3 and n = 3 thru 0 

8.  k=0, 0 pennies, 3 nickels, 1 dime = $0.25
9.  k=1, 5 pennies, 2 nickels, 1 dime = $0.25
10.  k=2, 10 pennies, 1 nickels, 1 dime = $0.25
11.  k=3, 15 pennies, 0 nickels, 1 dime = $0.25

For d = 2, we have

k + n = 1, there are 2 possibilities, k = 0 thru 1 and n = 1 thru 0

12.  k=0, 0 pennies, 1 nickel, 2 dimes = $0.25
13.  k=1, 5 pennies, 0 nickels, 2 dimes = $0.25

So the total is 1 + 6 + 4 + 2 = 13 ways.

Edwin
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