Question

Sum of three natural numbers a, b and c is 10. How many ordered triplets (a, b, c) exist?

Explanatory Answer

a + b + c = 10. Now, let us place ten sticks in a row

|       |       |       |       |       |       |       |       |       |

This question now becomes the equivalent of placing two '+' symbols somewhere between these sticks. For instance

|       |       |       |  +  |       |       |       |       |  +  |, 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 ⁹C₂. 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: ⁹C₂