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  4. The coefficient of x 49 in the product (x-1)(x-2)⋅s(x-50) is

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Q. The coefficient of $x ^{49}$ in the product $\left(x-1\right)\left(x-2\right)\cdots\left(x-50\right)$ is

KEAMKEAM 2014Binomial Theorem

Solution:

$(x-1)(x-2)(x-3) \ldots(x-50)$
$=x^{50}-(1+2+3+\ldots+50) x^{49}+(1 \cdot 2+2 \cdot 3+\ldots+49 \cdot 50) x^{48}+\ldots+(1 \times 2 \times 3 \times 4 \times \ldots \times 50)$
$\therefore $ Coefficient of $x^{49}=-(1+2+3+\ldots+50)$
$=-\left[\frac{50(50+1)}{2}\right] $
$=-25 \times 51 $
$=-1275$