The coefficient of a^{10} b^7 c^3 in the expa | vedclass

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(C) The general term in the expansion of $(bc + ca + ab)^{10}$ is given by the multinomial theorem as: $\frac{10!}{n_1! n_2! n_3!} (bc)^{n_1} (ca)^{n_

Vedclass · Accepted Answer

$120$

Vedclass · Answer

(C) The general term in the expansion of $(bc + ca + ab)^{10}$ is given by the multinomial theorem as: $\frac{10!}{n_1! n_2! n_3!} (bc)^{n_1} (ca)^{n_2} (ab)^{n_3} = \frac{10!}{n_1! n_2! n_3!} a^{n_2+n_3} b^{n_1+n_3} c^{n_1+n_2}$,where $n_1 + n_2 + n_3 = 10$.We need the coefficient of $a^{10} b^7 c^3$. Comparing the powers,we have:$n_2 + n_3 = 10$$n_1 + n_3 = 7$$n_1 + n_2 = 3$Adding these equations: $2(n_1 + n_2 + n_3) = 20$,which is consistent with $n_1 + n_2 + n_3 = 10$.Solving for $n_1, n_2, n_3$:From $n_1 + n_2 + n_3 = 10$ and $n_2 + n_3 = 10$,we get $n_1 = 0$.Substituting $n_1 = 0$ into $n_1 + n_2 = 3$,we get $n_2 = 3$.Substituting $n_2 = 3$ into $n_2 + n_3 = 10$,we get $n_3 = 7$.The coefficient is $\frac{10!}{0! 3! 7!} = \frac{10 	imes 9 	imes 8}{3 	imes 2 	imes 1} = 10 	imes 3 	imes 4 = 120$.

The coefficient of $a^{10} b^7 c^3$ in the expansion of $(bc + ca + ab)^{10}$ is

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