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
• B. 12376
• C. 9872
• D. 11752

Answer 1

The correct option is B 12376

We 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 = ${}^{(11+7-1)}{C}_{(7-1)}{=}^{17}{C}_6$