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Q.
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
Permutations and Combinations
Solution:
Let $b=a+d$ and $c=a+2 d$.
Given: $a+a+d+a+2 d=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 $\left(x+x^{2}+x^{3}+\ldots .\right)^{2}=$ Coefficient of $x^{12}$ in $\left(1+x+x^{2} \ldots\right)^{2}={ }^{13} C_{12}=13$.