Question

The number of nonnegative integer solutions of $x+y+2z$ $=$ $20$ is

• A. 76
• B. 84
• C. 112
• D. 121

Answer 1

Answer: (D)

121

This is equivalent to finding the coefficient of $x^{20}$ in $(1+x+x^2+x^3+...).(1+x+x^2+x^3+...).(1+x^2+x^4+x^6+...)={(1-x)}^{-2}.(1+x^2+x^4+x^6+...)=(1+2x+3x^2+4x^3+...21x^{20}).(1+x^2+x^4+x^6+...+x^{20})$
Thus, the answer is $=1+3+5+7+...+21={11}^2=121$
Hence, (D) is correct.