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Prove that sum(r=0)^n r(n-r)(^nC r)^2=n^...

Prove that `sum_(r=0)^n r(n-r)(^nC_ r)^2=n^2(^(2n-2)C_n)dot`

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Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .

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CENGAGE ENGLISH-BINOMIAL THEOREM-All Questions
  1. Prove that ^10 C1(x-1)^2-^(10)C2(x-2)^2+^(10)C3(x-3)^2+-^(10)C(10)(x-1...

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  2. If the third term in the expansion of (1+x)^mi s-1/8x^2, then find the...

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  3. Prove that sum(r=0)^n r(n-r)(^nC r)^2=n^2(^(2n-2)Cn)dot

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  4. Prove that 1-^n C1(1+x)/(1+n x)+^n C2(1+2x)/((1+n x)^2)-^n C3(1+3x)/((...

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  5. Find the coefficient of x^(20) in (x^2+2+1/(x^2))^(-5)(1+x^2)^(40)dot

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  6. The number of terms in the expansion of (a+b+c)^n , where n in Ndot

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  7. Find the coefficient of x^(50) in the expansion of (1+x)^(101)xx(1-x+x...

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  8. Find the coefficient of x^4 in the expansion of (2-x+3x^2)^6dot

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  9. Find the coefficient of x^k in1+(1+x)+(1+x)^2++(1+x)^n(0lt=klt=n)dot

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

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  11. If aa n db are distinct integers, prove that a-b is a factor of a^n-b^...

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  12. Find the a ,b ,a n dn in th expansion of (a+b)^n if the first three te...

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  13. Find the coefficient of x^(25) in expansion of expression sum(r=0)^(50...

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  14. If the sum of the coefficients of the first, second, and third terms ...

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  15. If p+q=1, then show that sum(r=0)^n r^2^n Crp^r q^(n-r)=n p q+n^2p^2do...

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  16. If (18 x^2+12 x+4)^n=a0+a(1x)+a2x2++a(2n)x^(2n), prove that ar=2^n3^r(...

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  17. Prove that ^m C1^n Cm-^m C2^(2n)Cm+^m C3^(3n)Cm-=(-1)^(m-1)n^mdot

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  18. Prove that "^n C0^(2n)Cn-^n C1^(2n-1)Cn+^n C2xx^(2n-2)Cn++(-1)^n^n Cn^...

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  19. Find the sum sum(r=0)^n^(n+r)Cr .

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  20. Find the value of sumsum(0leilejlt=n)(i+j)(nCi+n Cj)dot

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