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Prove the following identieties using th...

Prove the following identieties using the theory of permutation where `C_(0),C_(1),C_(2),……C_(n)` are the combinatorial coefficents in the expansion of `(1+x)^n,n in N:`
`""^(100)C_(10)+5.""^(100)C_(11)+10 .""^(100)C_(12)+ 10.""^(100)C_(13)+ 10.""^(100)C_(14)+ 10.""^(100)C_(15)=""^(105)C_(90)`

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If C_(0),C_(1),C_(2),…,C_(n) are the binomial coefficients in the expansion of (1+x)^(n)*n being even, then C_(0)+(C_(0)+C_(1))+(C_(0)+C_(1)+C_(2))+….+(C_(0)+C_(1)+C_(2)+…+C_(n-1)) is equal to :

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ALLEN-Solutions of Triangle & Binomial Theorem-EXERCISE (S-1)
  1. (2) If the coefficients of (2r + 4)th, (r - 2)th terms in the expansio...

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  2. Find the term independent of x in the expansion of [1/2x^(1//3)+x^(-1...

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  3. Prove that the ratio of the coefficient of x^(10) in the expansion of ...

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  4. underset(r=0)overset(n)(sum)(-1)^(r).^(n)C(r)[(1)/(2^(r))+(3^(r))/(2^(...

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  5. Find the numerically Greatest Term In the expansion of (3-5x)^15 when ...

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  6. Find the term independent of x in the expansion of (1+x+2x^3)[(3x^2//2...

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  7. Let (1+x^2)^2 . (1+x)^n = Sigma(k=0)^(n+4) (ak).x ^k "if" a1,a2 & a3...

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  8. Let f(x) = 1 - x +x^2-x^3+......+x^16+x^17 , then coefficient of x^2...

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  9. Let N=""^(2000)C1+2 .""^(2000)C2+3 .""^(2000)C(3)+....+2000.""^(2000)C...

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  10. Find the coefficient of x^2 y^3 z^4 in the expansion of (ax -by +cz...

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  11. Find the coefficient of x^4 in the expansion of (1+x+x^2 +x^3 )^11

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  12. The coefficient of x^r[0lt=rlt=(n-1)] in the expansion of (x+3)^(n-1)+...

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  13. If (1+x+x^2)6n=a0+a1x+a2x^2+……….=a(2n)x^(2n) then (A) a0+a3+a6+….=3^(n...

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  14. Prove the following identieties using the theory of permutation where ...

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  15. If C(0),C(1),C(2)…….,C(n) are the combinatorial coefficient in the exp...

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  16. Prove that C1/C0+(2c(2))/C1+(3C3)/(C2)+......+(n.Cn)/(C(n-1))=(n(n+1))...

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