Question

If $a,b,c$ are three natural numbers in $A.P.$ such that $a+b+c=21,$ then the possible number of values of $a,b,c$ is

• A. 13
• B. 14
• C. 15
• D. 16

Answer 1

The correct option is A 13

Let $b=a+d$ and $c=a+2d.$

Given: $a+a+d+a+2d=21\Rightarrow a+d=7$

$\therefore a+c=14$ and $b=7$

Now, the number of positive integral solutions of $a+c=14$

is equal to coefficient of $x^{14}$ in ${(x+x^2+x^3+...)}^2$

$=$ Coefficient of $x^{12}$ in ${(1+x+x^2...)}^2{=}^{13}{C}_{12}=13.$