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Prove that (""^(2n)C(0))^(2)-(""^(2n)C...

Prove that
`(""^(2n)C_(0))^(2)-(""^(2n)C_(1))^(2)+(""^(2n)C_(2))-(""^(2n)C_(3))^(2)+......+(""^(2n)C_(2n))^(2)=(-1)^(n)(""^(2n)C_(n))^2.`

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The correct Answer is:
`=(-1)^(n).""^(2n)C_(n)`
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VIKRAM PUBLICATION ( ANDHRA PUBLICATION)-BINOMIAL THEOREM-TETUAL EXERCISES (EXERCISE - 6(a)) III.
  1. If the coefficients of x^(9),x^(10),x^(11) in expansion of (1+x)^(n) a...

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  2. If 36, 84, 126 are three successive binomial coefficients in the expan...

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  3. Find the sum of the coefficients of x^(32) and x^(-18) in the expansio...

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  4. If P and Q are the sum of odd terms and the sum of even terms respecti...

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  5. If P and Q are the sum of odd terms and the sum of even terms respecti...

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  6. If the coefficients of 4 consecutive terms in the expansion of (1+x)^(...

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  7. Prove that (""^(2n)C(0))^(2)-(""^(2n)C(1))^(2)+(""^(2n)C(2))-(""^(2n...

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  8. Prove that (C(0)+C(1))(C(1)+C(2))(C(2)+C(3))....(C(n-1)+C(n))=((n+1)...

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  9. Find the term independent of x in (1+3x)^(n)(1+(1)/(3x))^(n).

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  10. Show that the middle term in the expansion of (1+x)^(2n)" is "(1.3.5. ...

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  11. If (1+3x-2x^(2))^(10)=a(0)+a(1)x+a(2)x^(2).+…+a(20)x^(20) then prove t...

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  12. If (1+3x-2x^(2))^(10)=a(0)+a(1)x+a(2)x^(2).+…+a(20)x^(20) then prove t...

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  13. If (3sqrt(3)+5)^(2n+1)=x" and "f=x-[x] where ([x] is the integral part...

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  14. If R, n are positive integers, n is odd, 0lt F lt1" and if "(5sqrt(5...

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  15. If R, n are positive integers, n is odd, 0lt F lt1" and if "(5sqrt(5...

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  16. Prove that : Suppose that n is a natural number and I, F are respecti...

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  17. Prove that : Suppose that n is a natural number and I, F are respecti...

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  18. Find the number of irrational terms in the expansion of (5^(1//6)+2^(1...

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