Question

Which of the is/are true for the coefficient of $x^2{y}^3{z}^r$ in the expansion of $(x-y+z{)}^{12}$?

• A. When $r=7$, then the coefficient is $-\frac{12!}{2!3!7!}$
• B. When $r=6$, then the coefficient is $-\frac{12!}{2!3!6!}$
• C. When $r=8$, then the coefficient is $-\frac{12!}{2!3!8!}$
• D. When $r=5$, then the coefficient is $-\frac{12!}{2!3!5!}$

Answer 1

The correct option is A When $r=7$, then the coefficient is $-\frac{12!}{2!3!7!}$

Given expansion $(x-y+z{)}^{12}$
Now, general term of the expansion
$=\frac{12!}{k_1!k_2!k_3!}{x}^{k_1}(-y{)}^{k_2}{z}^{k_3}$
Where, $k_1+k_2+k_3=12$
Now, from coefficient of $x^2{y}^3{z}^r$, we get
$k_1=2,k_2=3\Rightarrow k_3=7=r$

So, the required coefficient
$=\frac{12!}{2!3!7!}(-1{)}^3=-\frac{12!}{2!3!7!}$