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  4. Coefficient of xn-1 in the expansion of, (x+3)n+(x+3)n-1(x+2)+(x+3)n-2(x+2)2+ ldots . .+(x+2)n is :

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Q. Coefficient of $x^{n-1}$ in the expansion of, $(x+3)^n+(x+3)^{n-1}(x+2)+(x+3)^{n-2}(x+2)^2+\ldots . .+(x+2)^n$ is :

Binomial Theorem

Solution:

$(x+3)^n+(x+3)^{n-1}(x+2)+\ldots \ldots .+(x+2)^n=(x+3)^n\left(\frac{1-\left(\frac{x+2}{x+3}\right)^{n+1}}{1-\frac{x+2}{x+3}}\right)$
$=\left[(x+3)^{n+1}-(x+2)^{n+1}\right]$
Coefficient of $x^{n-1}={ }^{n+1} C_{n-1}(3)^2-{ }^{n+1} C_{n-1} \times 4$