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""^((2n + 1))C0 - ""^((2n+ 1))C1 + ""^((...

`""^((2n + 1))C_0 - ""^((2n+ 1))C_1 + ""^((2n + 1))C_2 - ……+""^((2n + 1))C_(2n) = `

A

`-1`

B

1

C

3

D

`-2`

Text Solution

Verified by Experts

The correct Answer is:
B
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""^((2n + 1))C_0 + ""^((2n+ 1))C_1 + ""^((2n + 1))C_2 + ……+""^((2n + 1))C_n =

""^(2n +1)C_0^2 -""^(2n+1)C_1^2 + ""^(2n+1)C_2^2 -…….- ""^(2n+1)C_(2n+1)^2 =

((1 + ""^nC_1 + ""^nC_2 + ""^nC_3+…….+nC_n)^2)/(1 + ""^(2n)C_1 + ""^(2n)C_2 + ""^(2n)C_3 + ……… + ""^(2n)C_(2n)) =

""^(2n)C_(n+1)+2. ""^(2n)C_(n) + ""^(2n) C_(n-1) =

Show that ""^(n)C_(0) +""^((n+1))C_(1) +""^((n+2))C_(2) +…+""^((n+k))C_(k) =""^((n+k+1))C_(k)

Prove that (C_(0) +C_(1)+C_(2)+….+C_(n))^(2)=1 +""^(2n)C_(1) +""^(2n)C_(2) +…..+""^(2n)C_(2n)

Prove that, C_(0) *""^(2n)C_(n)-C_(1)*""^((2n-2))C_(n)+C_(2) *""^((2n-4))C_(n)-…..=2^(n)

Prove that (""^(2n)C_(0))^(2)-(""^(2n)C_(1))^(2)+(""^(2n)C_(2))-(""^(2n)C_(3))^(2)+......+(""^(2n)C_(2n))^(2)=(-1)^(n)(""^(2n)C_(n))^2.

DIPTI PUBLICATION ( AP EAMET)-BINOMIAL THEOREM-EXERCISE 1B (BINOMIAL COEFFICIENTS)
  1. If n is odd then C0^2 - C1^2+C2^2-…....+(-1)^n Cn^2=

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  2. If n is even then C0^2 - C1^2+C2^2-…....+(-1)^n Cn^2=

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  3. ""^((2n + 1))C0 - ""^((2n+ 1))C1 + ""^((2n + 1))C2 - ……+""^((2n + 1))C...

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  4. Prove that (""^(2n)C(0))^(2)-(""^(2n)C(1))^(2)+(""^(2n)C(2))-(""^(2n...

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  5. C1//1-C2//2+C3//3-C4//4+…+(-1)^(n-1) Cn//n=

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  6. C0 - 2^3 . C1 +3^2. C2 - …. + (-1)^n (n+1)^2 . (Cn)=

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  7. Use the identity (1+x)^(m)(1+x)^(n)=(1+x)^(m+n) to prove Vandermonde's...

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  8. C0 - C2 + C4 - C6 +….....

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  9. C1 - C3 + C5 - C7 + …...

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  10. The term independent of x in (1+x)^n (1+(1)/(x))^n is

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  11. If a(n)=sum(r=0)^(n)(1)/(""^(n)C(r)) then sum(r=0)^(n)(r)/(""^(n)C(r))...

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  12. If x+y=1, then sum(r=6)^(n) r ^n Cr x^r . Y^(n-r)=

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  13. If x+y=1 then sum(r=0)^(n) r^2. ^(n) Cr x^r .y^(n-r)=

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  14. the value of (1)/((81)^n)-^(2n) C1. ((10)^2)/((81)^n)+ ^(2n) C1 ((10)^...

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  15. There are two bags each of which contains n balls. A main has to selec...

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  16. The sum of the coefficient of (1+2x-4x^2)^35 is

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  17. The sum of the coefficients of (5x-4y)^n where n is positive integer, ...

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  18. The sum of the coefficients in the expansion of (1+x-3x^2)^(171) is

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  19. The sum of the coefficients of even powers of x in the expansion of (1...

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  20. The sum of the coefficients of even powers of x in the expansion of (...

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