Question

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

Explanatory Answer

a + b + c = 10. a, b, c are whole numbers. Now this is similar to the previous question that we solved by placing 10 sticks and simplifying. The discussion can be seen here.

We cannot follow a similar approach as a, b, c can be zero. Let us modify the approach a little bit. Let us see if we can remove the constraint that a, b, c can be zero.

If we give a minimum of 1 to a, b, c then the original approach can be used. And then we can finally remove 1 from each of a, b, c. So, let us distribute 13 sticks across a, b and c and finally remove one from each.

a + b + c = 13. 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 + 4. or, a = 4, b = 5, c = 4.

There are 12 slots between the sticks, out of which one has to select 2 for placing the '+'s.

The number of ways of doing this is ¹²C₂.

Correct Answer: ¹²C₂