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  • The co-efficient of in the polynomial <img alt="(x-1)(x-2)(x-3)......(x-n)" src="https://learn.careers360.co

The co-efficient of x^{n-2} in the polynomial (x-1)(x-2)(x-3)......(x-n) is

Option: 1

\frac{n(n^{2}+2)(3n+1)}{24}
 


Option: 2

\frac{n(n^{2}-1)(3n+2)}{24}

 


Option: 3

\frac{n(n^{2}+1)(3n+4)}{24}


Option: 4

none of these


Answers (1)

best_answer

 

General Term in the expansion of (x+a)^n -

T_{r+1}= ^{n}c_{r}\cdot x^{n-r}\cdot a^{r}
 

- wherein

Where r\geqslant 0 \, and \, r\leqslant n

r= 0,1,2,----n

 

 

E=(x-\alpha _{1})(x-\alpha _{2})(x-\alpha _{3})........(x-\alpha _{n})

where \alpha _{1}=1,\alpha _{2}=2\; \; \; etc

                   =x^{n}-(\sum \alpha _{1})x^{n-1}+(\sum \alpha _{1}\alpha _{2})x^{n-2}+......

Hence co-efficient of x^{n-2}= sum of all the products of the first 'n' natural numbers taken two at a time

=\frac{(1+2+3+......n)^{2}-(1^{2}+2^{2}+.......n^{2})}{2}

=\frac{n(n^{2}-1)(3n+2)}{24}

 

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