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I need help solving this. I cannot find the complete number of combinations. I have already found $5$, but I can't find any more.

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    Which ones do you have? – 2017-02-28
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    1 quarter, 12 pennies – 2017-02-28
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    A quarter is $25$ cents, not $20$. – 2017-02-28
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    That's 37 cents ... – 2017-02-28
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    oops I forgot that they were 25 – 2017-02-28
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    Look for nonnegative integral solutions of the linear equation $p+5n+10d+25q=32$. – 2017-02-28
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    What the heck does that mean I am only a 4th grader – 2017-02-28

2 Answers 2

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First consider quarters. You could have either no quarters (leaving $32$ cents to be covered by pennies, nickels and dimes) or one quarter (leaving $7$ cents). Diagram:

     (32) 
  [0 quarters]        [1 quarter]         
     (32)                (7)

Next, consider dimes. In the first case you could have 0, 1 or 2 dimes, in the second you must have 0. Diagram:

     (32)
  [0 quarters]                              [1 quarter]         
     (32)                                      (7) 
[0 dimes]    [1 dime]    [2 dimes]          [0 dimes]
  (32)         (22)         (2)                (7)

Next consider nickels. Finally, everything left over must be done with pennies.

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Leaving out the pennies in each combo ...

There are 2 combos with a quarter:

25+5 (that is: 1 quarter + 1 nickel ...so 2 pennies...)

25 (So just a quarter ...so 7 pennies ... for combos below, you'll have to figure out how many pennies to add ...)

There is 1 combo with 3 dimes:

10+10+10

There are 3 combos with 2 dimes:

10+10+5+5

10+10+5

10+10

There are 5 combos with 1 dime:

10+5+5+5+5

10+5+5+5

10+5+5

10+5

10

There are 7 combos without dimes or quarters:

5+5+5+5+5+5

5+5+5+5+5

5+5+5+5

5+5+5

5+5

5

(32 pennies)

Total: 18 combos

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    thanks for your help @Bram28 – 2017-02-28
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    @SagePosko No problem ... do you see the way I created these combos so we can be sure there are no others? – 2017-02-28
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    yes I do, now I know what to do next time – 2017-02-28
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    can you help with another question?@Bram28 – 2017-02-28
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    @SagePosko sorry, just saw your question. I would say if you have another question then just make that another post ... There are lots of people in the community that can help! :) – 2017-03-01