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The question is from Permutation and Combination. It is about counting natural numbers. We need to find out the ordered triplets given the sum of three natural numbers. This section hosts a number of questions which are on par with CAT questions in difficulty on CAT Permutation and Combination, and CAT Probability.

Question 1: Sum of three Natural numbers a, b and c is 10. How many ordered triplets (a, b, c) exist?

  1. 45
  2. 36
  3. 54
  4. 28

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Explanatory Answer

Method of solving this CAT Question from Permutation and Combination: One of our facvourite questions. One of those that get simplified if we think about it the right way, but is extremely tough if you do not get the right line of thought

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.

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

Hence the answer is "9C2 = 36."

Choice B is the correct answer.


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