Question

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

• A. - 2280
• B. - 1275
• C. 1275
• D. -2250
• E. - 49

Answer 1

Answer: (B)

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