Binomial Theorem

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P.T:

(n c 0) (2n c n) - (n c 1) (2n-1 c n) +....

[(-1)^n] (n c n) (n c n)] =8

#Mathematics #Algebra

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Manmay kumar Mohanty ·2010-04-05 00:21:37

u got the question rong .......
we have to prove if L.H.S. = 1 not 8

²ⁿC_n is coefficient of xⁿ in expansion of (1+x)²ⁿ
^2n-1C_n is coefficient of xⁿ in expansion of (1+x)^2n-1
^2n-2C_n is coefficient of xⁿ in expansion of (1+x)^2n-2

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ⁿC_n is coefficient of xⁿ in expansion of (1+x)ⁿ

we have L.H.S = C₀.²ⁿC_n-C₁^2n-1C_n- ........ (-1)ⁿ C_nⁿC_n
= coefficient of xⁿ in [C₀(1+x)²ⁿ - C₁(1+x)^2n-1 + ....+(-1)ⁿC_n(1+x)ⁿ

= coefficient of xⁿ in(1+x)ⁿ [C₀(1+x)ⁿ - C₁(1+x)^n-1 + ....+(-1)ⁿC_n]

= coefficient of xⁿ in(1+x)ⁿ [(1+x)-1]ⁿ
= coefficient of xⁿ in(1+x)ⁿ xⁿ
= 1
= R.H.S

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Binomial Theorem

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