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Prove that ^n C0 .^n C0-^(n+1)C1 . ^n C1...

Prove that `^n C_0 .^n C_0-^(n+1)C_1 . ^n C_1+^(n+2)C_2 . ^n C_2-=(-1)^ndot`

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If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .

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CENGAGE ENGLISH-BINOMIAL THEOREM-All Questions
  1. Find the fourth term in the expansion of (1-2x)^(3//2)""dot

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  2. Prove that .^n C0 . ^(2n) Cn- ^n C1 . ^(2n-2)Cn+^n C2 . ^(2n-4)Cn-=2...

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  3. Prove that ^n C0 .^n C0-^(n+1)C1 . ^n C1+^(n+2)C2 . ^n C2-=(-1)^ndot

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  4. Find the sum of the coefficients of all the integral powers of x in th...

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  5. If the sum of the coefficient in the expansion of (alpha^2x^2-2alphax+...

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  6. Prove that sum(alpha+beta+gamma = 10) (10 !)/(alpha!beta!gamma!)=3^(10...

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  7. If (1+x-2x^2)^(20)=a0+a1x+a2x^2+a3x^3+...+a(40)x^(40), then find the v...

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  8. Find the sum of the series .^15 C0+^(15)C1+^(15)C2+...............+^(1...

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  9. Find the sum sum(k=0)^(10).^(20)Ck.

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  10. Find the sum of all the coefficients in the binomial expansion of (x^2...

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  11. If the sum of coefficient of first half terms in the expansion of (x+y...

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  12. Find the value of sum(p=1)^n(sum(m=p)^n.^nCm.^m Cp)dot And hence, find...

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  13. Show that the middle term in the expansion of (1+x)^(2n)i s((1. 3. 5 (...

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  14. If the middle term in the expansion of (x^2+1//x)^n is 924 x^6 , then...

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  15. The first three terms in the expansion of (1+a x)^n(n!=0) are 1,6xa n ...

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  16. If x^4 occurs in the rth term in the expansion of (x^4+1/(x^3))^(15), ...

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  17. Find the coefficient of x^(-10) in the expansion of (a/x+b x)^(12) .

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  18. Find the constant term in the expansion of (x-1//x)^6dot

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  19. If the coefficients of (r-5)t h a n d(2r-1)t h terms in the expansion ...

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  20. In ( 2^(1/3)+1/3^(1/ 3))^n if the ratio of 7th term from the beginnin...

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