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Prove that ^n C0^(2n)Cn-^n C1^(2n-1)Cn+^...

Prove that `^n C_0^(2n)C_n-^n C_1^(2n-1)C_n+^n C_2xx^(2n-2)C_n++(-1)^n^n C_n^n C_n=1.`

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`LHS = C_(0) * ""^(2n)C_(n) - C_(1) * ""^(2n-1)C_(n) + C_(2) . ""^(2n-4)C_(n) - ...` ltbrge Coefficient of ` x^(n)` in
` [C_(0)(1 + x)^(2n) - C_(1) (1 + x)^(2n-2) + C_(2) (1 + x)^(2n-4)-...]`
= Coefficient of ` x^(n) ` in
`[C_(0) (1 + x)^(2)]^(n) C_(1) [ (1 + x)^(n) ]^(n-1) = C_(2) [(1 + x)^(2)] ^(n-2) - ...]`
= Coefficient of ` x^(n) " in" (1 x)^(n) [((1 +x)-1)^(n)]`
= Coefficient of ` x^(n) "in" (1 + x)^(n) * x^(n) d `
= Constant term in ` (1 +x)^(n) = 1 = RHS `
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ARIHANT MATHS ENGLISH-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that ^n C0^(2n)Cn-^n C1^(2n-1)Cn+^n C2xx^(2n-2)Cn++(-1)^n^n Cn^n...

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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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