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

Answer: (A)

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 ${\left(x+x^2+x^3+\dots .\right)}^2=$ Coefficient of $x^{12}$ in ${\left(1+x+x^2\dots \right)}^2={}^{13}{C}_{12}=13$.