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  4. If coefficient of x49 in the product (x - 1) (x - 2) (x-3).......... (x - 50) is

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Q. If coefficient of $x^{49}\,$ in the product $ (x - 1) (x - 2) (x-3).......... (x - 50)$ is

Binomial Theorem

Solution:

$(x - 1) (x - 2) (x - 3 ) .......... (x - 10)$
$= x^{50} - x^{49} (1 + 2 + 3 + ......... + 10) + .......$
$\therefore $ co.eff of of $x^{49} = - (1 + 2 + ...... + 10)$
$= - \frac{50\left(50\,1\right)}{2}$
$= - 25 \times 51 = -1275$