Question

If $a,b,c$ are three natural number 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 C $15$

Let $b=a+d$, $c=a+2d$

Given $a+a+d+a+2d=21$ so $a+d=7$

Thus $a+c=14$ and $b=7$

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

$={}^{n+r-1}{C}_{r-1}$
$={}^{14+2-1}{C}_{2-1}$
$={}^{15}{C}_1=15$