The coefficient of x^5 in the expansion of (x | vedclass

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(A) The general term in the multinomial expansion of $(3+2x+x^2)^5$ is given by $\frac{5!}{p!q!r!} (3)^p (2x)^q (x^2)^r$,where $p+q+r=5$.We need the c

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$1052$

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(A) The general term in the multinomial expansion of $(3+2x+x^2)^5$ is given by $\frac{5!}{p!q!r!} (3)^p (2x)^q (x^2)^r$,where $p+q+r=5$.We need the coefficient of $x^5$,so $q+2r=5$.Possible non-negative integer solutions for $(p, q, r)$ are:$1$) If $r=0$,then $q=5$,$p=0$. Term: $\frac{5!}{0!5!0!} (3)^0 (2)^5 (1)^0 = 1 	imes 1 	imes 32 	imes 1 = 32$.$2$) If $r=1$,then $q=3$,$p=1$. Term: $\frac{5!}{1!3!1!} (3)^1 (2)^3 (1)^1 = 20 	imes 3 	imes 8 = 480$.$3$) If $r=2$,then $q=1$,$p=2$. Term: $\frac{5!}{2!1!2!} (3)^2 (2)^1 (1)^2 = 30 	imes 9 	imes 2 = 540$.Summing these coefficients: $32 + 480 + 540 = 1052$.

The coefficient of $x^5$ in the expansion of $(x^2+2x+3)^5$ is

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