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Use a combinatorial argument to prove th...

Use a combinatorial argument to prove that `(C(n,1))^2+2(C(n,2))^2+3(C(n,3))^2+...........+n(C(n,n))^2=((2n-1)!)/(((n-1)!)^2)`

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A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Prove that sum(r=1)^k (-3)^(r-1) "^(3n)C(2r-1) =0 , where k=(3n)/2 and...

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  2. 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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  3. Use a combinatorial argument to prove that (C(n,1))^2+2(C(n,2))^2+3(C...

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  4. Prove that (""^nC1)/(""^nC0)+2*(""^nC2)/(""^nC1)+3*(""^nC3)/(""^nC2)+....

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  5. Given, sn=1+q+q^2++q^n ,Sn=1+(q+1)/2+((q+1)/2)^2++((q+1)/2)^n ,q!=1 p...

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  6. Find the value of sum(p=1)^n(sum(m=p)^n^n Cm^m Cp)dot And hence, find ...

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  7. The value of ^(2n+1)C0^2+^(2n+1)C1^2+^(2n+1)C2^2+....+^(2n+1)Cn^2 is e...

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  8. Find the sum sumsum(0lt=i<jlt=n)^n Ci^n Cj

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  9. If (1 + x + x^(2) + x^(3))^(n) = a(0) + a(1)x + a(2)x^(2)+"……….."a(3n)...

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  10. The coefficient of a^4b^5 in the expansion of (a+b)^9 is .

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  11. The coefficient in the third term of the expansion of (x^2-1/4)^n when...

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  12. Which is larger : (99^(50)+100^(50)) or (101)^(50).

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  13. The sum of the coefficeints of the polynominal (1 + x - 3x^(2))^(2163)...

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  14. The sum of the numerical coefficients in the expansion of (2x+3y)^10 i...

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  15. If the fifth term of the expansion (a^(2//3)+a^(-1))^n does not contai...

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  16. The sum of the rational terms in the expansion of (sqrt(2)+ root(5...

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  17. Write the middle term in the expansion of (x+1/x)^(10)dot

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  18. If the coefficients of 2nd, 3rd and 4th terms in the expansion of(1+x)...

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  19. If x^r occurs in the expansion of (x+1/x)^n then its coefficient is .

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  20. If the sum of the coefficients in the expansion of (a+b)^n is 4096, th...

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