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

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

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

(*^(n)c_(0))^(2)+3(.^(n)C_(1))^(2)+5(.^(n)C_(2))^(2)+ ......+(2n+1)(.^(n)C_(n))^(2)

The value of sum (""^(n) C_(1) )^(2)+ (""^(n) C_(2) )^(2) + (""^(n) C_(3))^(2) + …+ (""^(n) C_(n) )^(2) is

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-2)C_(n)+^(n)C_(2)^(2n-4)C_(n)-...=2^(n)

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

Prove that : ""^(n)C_(0).""^(2n)C_(n)-""^(n)C_(1).""^(2n-2)Cn_(n)+""^(n)C_(2).""^(2n-4)Cn_(n)+......=2^n

Prove that ^nC_(0)^(n)C_(0)-^(n+1)C_(1)^(n)C_(1)+^(n+2)C_(2)^(n)C_(2)-...=(-1)^(n)

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. Find sum(r=0)^n(r+1)*"^nCrx^r

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  2. Show that C0^2-C1^2+C2^2-C3^2+...........+(-1)^n Cn^2=0 or (-1)^(n/2)C...

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

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  4. Sum of the products of the binomial coefficients C0,C1,C2,......Cn ta...

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  5. Find the sum ^20C10.^15C0+^20C9.^15C1+^20C8.^15C2+....+^20C0.^15C10

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  6. Prove that sum(r=1)^k (-3)^(r-1) "^(3n)C(2r-1) =0 , where k=(3n)/2 and...

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

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

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

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

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

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

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

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

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

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

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

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