Question

In the expansion of $(x+y+z{)}^{25}$

• A. every term is of the form ${}^{25}{\text{C}}_r{\cdot }^r{\text{C}}_k\cdot x^{25-r}\cdot y^{r-k}.z^k$
• B. the coefficient of $x^8{y}^9{z}^9$ is $0$
• C. the number of terms is $325$
• D. none of these

Answer 1

Answer: (A), (B)

the coefficient of $x^8{y}^9{z}^9$ is $0$
every term is of the form ${}^{25}{\text{C}}_r{\cdot }^r{\text{C}}_k\cdot x^{25-r}\cdot y^{r-k}.z^k$

In the above expansion we apply the multinomial theorem.
The general term will be of the form
${}^{25}{C}_r{.}^r{C}_k{x}^{25-r}{y}^{r-k}{z}^k$.
Note that in the above expression of the general term, the sum of the powers of $x,y,z$ is always equal to $25$.
Consider the term $x^8{y}^9{z}^9$ since the sum of the powers of $x,y,z$ is $8+9+9$
$=26$
Hence, the term will not exist as sum of the powers of the $x,y,z$ is greater than $25$.