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

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))`.

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Verified by Experts

`LHS = (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1))`
` prod _(r=1)^(n) (C_(r-1) +2 ""^(n)C_(r) + C_(r+1))`
` prod _(r=1)^(n) {(""^(n)C_(r-1) +""^(n)C_(r))+ (""^(n)C_(r)+ ""^(n)C_(r-1))}`
` prod _(r=1)^(n) (""^(n+1)C_(r-1) + ""^(n+1)C_(r+1))" "` [by Pascal's rule]
` prod _(r=1)^(n) (""^(n+2)C_(r-1) )= prod_(r=1)^(n) ((n+2)/(r+1)) ""^(n+1)C_(r)[because ""^(n)C_(r)= n/r*""^(n-1)C_(r-1)]`
` prod _(r=1)^(n) ((n+2)/(r+1))(""^(n+2)C_(r-1) )= prod_(r=1)^(n) ((n+2)/(r+1))prod_(r=1)^(n) (C_(r-1)+ C_(r))`
` = ((n+2))/(2) *((n+2))/(3) *((n+2))/(4) ...((n+2))/((n+1)) prod_(r=1)^(n) (C_(r-1) + C_(r))`
`((n+2)^(n))/((n+1)!) prod_(r=1)^(n) (C_(r-1) + C_(r))= RHS `
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ARIHANT MATHS-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
  1. If C(0) , C(1), C(2), …, C(n) are the binomial coefficients in the...

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

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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

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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)^4(1+x^3)^7(1+x^4)^(1...

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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)))^n x ne 0,...

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

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