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If C(0), C(1), C(2),…, C(n) denote the b...

If `C_(0), C_(1), C_(2),…, C_(n)` denote the binomial
coefficients in the expansion of `(1 + x)^(n) ,` then `sum_(r=0)^(n)sum_(s=0)^(n)(C_(r) +C_(s))`

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To solve the problem, we need to evaluate the expression: \[ \sum_{r=0}^{n} \sum_{s=0}^{n} (C_r + C_s) \] where \(C_r\) and \(C_s\) are the binomial coefficients from the expansion of \((1 + x)^n\). ...
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If C_(0), C_(1), C_(2), …, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then C_(0)""^(2) + 2 C_(1)""^(2) + 3C_(2)""^(2) + ...+ (n +1)C_(n)""^(2) =

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If C_(0), C_(1), C_(2),...,C_(n) denote the binomial coefficients in the expansion of (1 + x)^n) , then xC_(0)-(x -1) C_(1)+(x-2)C_(2)-(x -3)C_(3)+...+(-1)^(n) (x -n) C_(n)=

If C_(0), C_(1), C_(2), ..., C_(n) denote the binomial cefficients in the expansion of (1 + x )^(n) , then a C_(0) + (a + b) C_(1) + (a + 2b) C_(2) + ...+ (a + nb)C_(n) = .

If C_(0), C_(1), C_(2), ..., C_(n) denote the binomial cefficients in the expansion of (1 + x )^(n) , then a C_(0) + (a + b) C_(1) + (a + 2b) C_(2) + ...+ (a + nb)C_(n) = .

If C_(0),C_(1), C_(2),...,C_(N) denote the binomial coefficients in the expansion of (1 + x)^(n) , then 1^(3). C_(1)-2^(3). C_(3) - 4^(3) . C_(4) + ...+ (-1)^(n-1)n^(3) C_(n)=

If C_(0), C_(1), C_(2),..., C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then . 1. C_(1) - 2 . C_(2) + 3.C_(3) - 4. C_(4) + ...+ (-1)^(n-1) nC_(n)=

If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

If C_(0), C_(1) C_(2) ….., denote the binomial coefficients in the expansion of (1 + x)^(n) , then (C_(0))/(2) - (C_(1))/(3) + (C_(2))/(4)- (C_(3))/(5)+...+ (-1)^(n)(C_(n))/(n+2) =

ARIHANT MATHS ENGLISH-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
  1. If C(0), C(1), C(2),…, C(n) denote the binomial coefficients in the...

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  2. The value of ((30), (0))((30), (10))-((30), (1))((30),( 11)) +(30 2)(3...

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  3. If the coefficient of the rth, (r+1)th and (r+2)th terms in the expans...

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  4. If the coefficient of x^(7)in [ax^(2) + (1/bx)]^(11) equals the coeffi...

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  5. For natural numbers m ,n ,if(1-y)^m(1+y)^n=1+a1y+a2y^2+... , and a1=a2...

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  6. In the binomial expansion of (a - b)^n , n ge 5 the sum of the 5th ...

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  7. The sum of series ^^(20)C0-^^(20)C1+^^(20)C2-^^(20)C3++^^(20)C 10 is 1...

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  8. Statement-1: sum(r =0)^(n) (r +1)""^(n)C(r) = (n +2) 2^(n-1) Stat...

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  9. The reamainder left out when 8^(2n) - (62)^(2n+1) is divided by 9 is

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  10. For r = 0, 1,"…..",10, let A(r),B(r), and C(r) denote, respectively, t...

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  11. Let S(1) = sum(j=1)^(10) j(j-1).""^(10)C(j), S(2) = sum(j=1)^(10)j."...

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  12. Find the coefficient of x^7 in the expansion of (1 - x -x^2 + x^3)^(6)...

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  13. If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is ...

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  14. The term independent of x in expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(...

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  15. The coefficients of three consecutive terms of (1+x)^(n+5) are in the ...

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  16. If the coefficient of x^(3) and x^(4) in the expansion of (1+ax+bx^(2)...

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  17. Coefficient of x^(11) in the expansion of (1+x^2)(1+x^3)^7(1+x^4)^(12)...

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  18. The sum of coefficient of integral powers of x in the binomial expansi...

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  19. The coefficient of x^9 in the expansion of (1+x)(16 x^2)(1+x^3)(1+x^(1...

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  20. If the number of terms in the expansion of (1-2/x+4/(x^(2))) x ne 0, i...

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  21. Let m be the smallest positive integer such that the coefficient of x^...

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