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Binomial Coefficients

Find the coef...

Question

Find the coefficient of $x^{8} i n (1 + x)^{8} (1 - x)^{2}$ .

A

= + + + .............+

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B

= + 2 + 3 + .............+ 9

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C

= + 2 + 2 + .............+ 2

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D

= + 3 + 3 + .............+ 3

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Solution

The correct option is B
= + 2 + 3 + .............+ 9

Coefficient of $x^{8}$ in $(1 + x)^{8} (1 - x)^{- 2}$

Write the binomial expansion of $(1 + x)^{8} (1 - x)^{- 2}$ .

= coefficient of $x^{8}$ in ( $^{8} C_{0} +^{8} C_{1} x +^{8} C_{2} x^{2} + . . . . . . . . . . . . . +^{8} C_{8} x^{8})$

× $(1 + 2 x + 3 x^{2} + 4 x^{3} + . . . . . . . . . . . . . . + 9 x^{8} + . . . . . . . . . . . . . .)$

= 9. $^{8} C_{0}$ + 8. $^{8} C_{1}$ + 7. $^{8} C_{2}$ + .................+ 1. $^{8} C_{8}$

= $^{8} C_{0}$ + 2 $^{8} C_{1}$ + 3 $^{8} C_{2}$ +.............+ 9 $^{8} C_{8}$

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