Question:
The coefficient of $x ^{49}$ in the product $\left(x-1\right)\left(x-2\right)\cdots\left(x-50\right)$ is
The coefficient of $x ^{49}$ in the product $\left(x-1\right)\left(x-2\right)\cdots\left(x-50\right)$ is
Updated On: Apr 8, 2024
- -2280
- -1275
- 1275
- -2250
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The Correct Option is B
Solution and Explanation
$(x-1)(x-2)(x-3) \ldots(x-50)$
$=x^{50}-(1+2+3+\ldots+50) x^{49}+(1 \cdot 2+2 \cdot 3+\ldots+49 \cdot 50) x^{48}+\ldots+(1 \times 2 \times 3 \times 4 \times \ldots \times 50)$
$\therefore$ Coefficient of $x^{49}=-(1+2+3+\ldots+50)$
$=-\left[\frac{50(50+1)}{2}\right] $
$=-25 \times 51 $
$=-1275$
$=x^{50}-(1+2+3+\ldots+50) x^{49}+(1 \cdot 2+2 \cdot 3+\ldots+49 \cdot 50) x^{48}+\ldots+(1 \times 2 \times 3 \times 4 \times \ldots \times 50)$
$\therefore$ Coefficient of $x^{49}=-(1+2+3+\ldots+50)$
$=-\left[\frac{50(50+1)}{2}\right] $
$=-25 \times 51 $
$=-1275$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
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- The first and the last terms are xn and yn respectively.
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