Question

Assertion :Number of terms in the expansion of $(x+y+z+w{)}^{50}$ is ${}^{53}{C}_3$. Reason: Number of non-negative solution of the equation $p+q+r+s=50$ is ${}^{53}{C}_3$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

General term in the expansion of $(x+y+z+w{)}^{50}$ is
$\frac{50!}{p!q!r!s!}{x}^p{y}^q{z}^r{w}^s$
where $p+q+r+s=50,0\le p,q,r,s\le 50$.
Now the number of terms is equal to number of ways in which we can adjust powers of x, y, z, and w such that their sum is 50, i.e., equal to the non-negative solutions of $p+q+r+s=50$, which is given by ${}^{50+4-1}{C}_{4-1}{=}^{53}{C}_3$.