The coefficient of a^{3} b^{4} c^{5} in the e | vedclass

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(C) The general term in the expansion of $(bc + ca + ab)^{6}$ is given by the multinomial theorem as $\frac{6!}{p! q! r!} (bc)^{p} (ca)^{q} (ab)^{r} =

Vedclass · Accepted Answer

$60$

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(C) The general term in the expansion of $(bc + ca + ab)^{6}$ is given by the multinomial theorem as $\frac{6!}{p! q! r!} (bc)^{p} (ca)^{q} (ab)^{r} = \frac{6!}{p! q! r!} a^{q+r} b^{p+r} c^{p+q}$.We need the coefficient of $a^{3} b^{4} c^{5}$,so we set the exponents equal to the required powers:$q + r = 3$$p + r = 4$$p + q = 5$Adding these three equations: $2(p + q + r) = 3 + 4 + 5 = 12$,so $p + q + r = 6$.Subtracting each equation from the sum:$p = (p + q + r) - (q + r) = 6 - 3 = 3$$q = (p + q + r) - (p + r) = 6 - 4 = 2$$r = (p + q + r) - (p + q) = 6 - 5 = 1$The coefficient is $\frac{6!}{3! 2! 1!} = \frac{720}{6 	imes 2} = \frac{720}{12} = 60$.

The coefficient of $a^{3} b^{4} c^{5}$ in the expansion of $(bc + ca + ab)^{6}$ is

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