Question

The number of solutions to $x+y+z=10$ , where $1\leq x,y,z\leq 6$ and $x,y,z\in N$ , is equal to

• A. $35$
• B. $36$
• C. $27$
• D. $66$

Answer 1

Answer: (C)

$x+y+z=10$
Number of solutions is equal to the coefficient of $x^{10}$ in the expansion of
$\left(x+x^{2}+x^{3}+\ldots \ldots+x^{6}\right)^{3}=x^{3}\left(\frac{1-x^{6}}{1-x}\right)^{3}$or the coefficient of $x^{7}$ in the expansion of $\left(1-x^{6}\right)^{3}(1-x)^{-3}=\left(1-3 x^{6}\right)(1-x)^{-3}$
$={ }^{7+3-1} C_{7}-3\left({ }^{1+3-1} C_{1}\right)$
$={ }^{9} C_{7}-3\left({ }^{3} C_{1}\right)=36-9=27$