Question
Sum of three whole numbers a, b and c is 10. How many ordered triplets (a, b,
c) exist? (This is a variant on the previous question that can be found
here). Will post the solution in a day or two.
Correct Answer
12C2
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.