Question

Find the number of ways in which we can get a sum less than or equal to 17 by adding six natural numbers.

A
13724
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B
12376
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C
9872
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D
11752
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Solution

The correct option is D 12376We are asked to find the number of ways in which 6 natural numbers can give a sum less than or equal to 17 means, a + b + c + d + e + f < = 17 a , b, c ... f can take values from 1. We know the direct formula: a + b + c ... k terms = n. non negative integral solutions = (n+k−1)C(k−1) But here is a catch. This holds for "Equal to n" and not for "Less than or Equal to n". To solve this, add a dummy variable. (say, g) so we have a + b + c + d + e + f + g = 17 One important thing here is than minimum value of a, b, c ... f is 1 (not 0) so our equation reduces to a + b + c + d + e + f + g = 17 - 6 = 11 Now our formula holds good and number of positive solution = (11+7−1)C(7−1)=17C6

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