Question

The number of solutions to $x+y+z=10$ , where $1\le x,y,z\le 6$ 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
${\left(x+x^2+x^3+\dots \dots +x^6\right)}^3=x^3{\left(\frac{1-x^6}{1-x}\right)}^3$or the coefficient of $x^7$ in the expansion of ${\left(1-x^6\right)}^3(1-x{)}^{-3}=\left(1-3x^6\right)(1-x{)}^{-3}$
$={}^{7+3-1}{C}_7-3\left({}^{1+3-1}{C}_1\right)$
$={}^9{C}_7-3\left({}^3{C}_1\right)=36-9=27$