Question

The number of solutions of the equation $x+y+z=10$ where $x,y$ and $z$ are positive integers

• A. $36$
• B. $55$
• C. $72$
• D. $45$

Answer 1

The correct option is A $36$

Given equation is $x+y+z=10$
where $x,y$ and $z$ are positive integers

For any pair of positive integers n and k, the number of k-tuples of positive integers whose sum is n is equal to the number of $(k-1)$-element subsets of a set with $(n-1)$ elements.

Both of these numbers are given by the binomial coefficient $(\frac{n-1}{k-1}).$

$\therefore$ Required number of solutions $={{}^{(10-1)}C}_{(3-1)}={{}^{9}C}_2=\frac{9\times 8}{2}=36$