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

Question

Infinity Learn · Accepted Answer

Let b $$=$$ a $$+$$ d and c $$=$$ a $$+$$ $$2d$$.Given: a $$+$$ a $$+$$ d $$+$$ a $$+$$ $$2d$$ $$=$$ $$21$$ ⇒ a $$+$$ d $$=$$ $$7$$∴ a $$+$$ c $$=$$ $$14$$ and b $$=$$ $$7Now$$, the number of positive integral solutions of a $$+$$ c $$=$$ $$14$$ is equal to coefficient of $$x14$$ in $$(x$$ $$+$$ $$x2$$ $$+$$ $$x3$$ $$+$$….$$)2$$ $$=$$ Coefficient of $$x12$$ in $$(1$$ $$+$$ x $$+$$ $$x2$$ …$$)2$$ $$=$$ $$13C12$$ $$=$$ $$13$$.

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

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