a + b + c = 10. Now, let us place ten sticks in a row
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This question now becomes the equivalent of placing two '+' symbols somewhere between these sticks. For instance
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This would be the equivalent of 4 + 5 + 1. or, a = 4, b = 5, c = 1.
There are 9 slots between the sticks, out of which one has to select 2 for placing the '+'s.
The number of ways of doing this would be ^{9}C_{2}. Bear in mind that this kind of calculation counts ordered triplets. (4, 5, 1) and (1, 4, 5) will both be counted as distinct possibilities.
We can also do a + b + c = n where a, b, c have to be whole numbers (instead of natural numbers as in this question) with a small change to the above approach. Give it some thought.
Correct Answer: ^{9}C_{2} = 36.