Question

Assertion :Coefficent of $a{b}^8{c}^3{d}^2$ in the expansion of ${(a+b+c+d)}^{14}$ is 180180 Reason: General term in the expansion of ${(a_1+a_2+a_3+...+a_m)}^n$
$=\sum \frac{n!}{n_1!n_2!n_3!....n_m!}{a}_1^{n_1}{a}_2^{n_2}...a_m^{n_m}$ where $n_1+n_2+n_3+...+n_m=n.$

• 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. Both Assertion and Reason are incorrect

Answer 1

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

$(x_1+x_2+x_3+...x+r{)}^n=\sum \frac{n!(x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}{x}_3^{{\alpha }_3}...x_r^{{\alpha }_r})}{{\alpha }_1!{\alpha }_2!{\alpha }_3!...{\alpha }_r!}$

Where ${\alpha }_1,{\alpha }_2,{\alpha }_3...{\alpha }_r$ are all non-negative integers and subjected to a condition that ${\alpha }_1+{\alpha }_2+{\alpha }_3+...{\alpha }_r=n$

Consider the multinomial theorem,

The coefficient of $a.b^8{c}^3.d^2$ in the expansion of $(a+b+c+d{)}^{14}$

$=\frac{14!}{1!.8!.3!.2!}$

$=180180$.