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Prove that 3*""^10C0+3^2*(""^10C1)/2+3^3...

Prove that `3*""^10C_0+3^2*(""^10C_1)/2+3^3*(""^10C_2)/3+...3^11*(""^10C_10)/11=(4^11-1)/11`

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Evaluate : ""^10C_1 + ""^10C_2 + ""^10C_3 + …. + ""^10C_10

Evaluate ""^(10)C_1 + ""^(10)C_2 + ""^(10)C_3 + ………+""^10C_10

Evaluate : 2^(10)C_(0)+(2^(2).^(10)C_(1))/(2)+(2^(3).^(10)C_(2))/(3)+ . . .+(2^(11).^(10)C_(10))/(11)

Prove that ^10C_(1)(x-1)^(2)-^(10)C_(2)(x-2)^(2)+^(10)C_(3)(x-3)^(2)+...-^(10)C_(10)(x-10)^(2)=

If (^(10)C_(0))/1+(^(10)C_(1))/2+(^(10)C_(2))/3+.....+(^(10)C_(10))/(11)=(2^(P)-1)/(q) then the value of (p)/(2q) is

(11C_(0))/(1)+(11C_(1))/(2)+(11C_(2))/(3)+......+(11C_(10))/(11)=

Prove that : ""^(25)C_(10)+""^(24)C_(10)+……..+""^(10)C_(10)=""^(26)C_(11)

Sum of the series S = 3^(-1)(""^(10)C_(0))-""^(10)C_(1)+(3)(""^(10)C_(2))-3^(2)(""^(10)C_(3))+…+3^(9)(""^(10)C_(10)) is

(.1^(11)C_(0))/(1)+(.^(11)C_(1))/(2)+(.^(11)C_(2))/(3)+....+(.1^(11)C_(10))/(11)

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that 2.C0+2^2C1/2+2...

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  2. If (1+x)^n=C0+C1x+C2x^2+C3x^3+...+Cnx^n then prove that C0-1/2C1+1/3C2...

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  3. Prove that 3*""^10C0+3^2*(""^10C1)/2+3^3*(""^10C2)/3+...3^11*(""^10C10...

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  4. Prove that 2*""^nC0+2^2*(""^nC1)/2+2^3*(""^nC2)/3+...2^(n+1)*(""^nCn)/...

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  5. Find the sum sum(k=0)^n("^nCk)/((k+1)(k+2))

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  6. Find the sum sum(r=0)^n(-1)^r*(""^nCr)/(""^(r+3)Cr)

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  7. Find the sum sum(k=1) (""^nC(2k-1))/(2k)

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  8. Find the sum sum(k=0)^n ("^nCk)/(k+1)

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  9. If (1+x)^n=sum(r=0)^n Crx^r then prove that sum(r=0)^n (Cr)/((r+1)2^(r...

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  10. Find sum(r=0)^n(r+1)*"^nCrx^r

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

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

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

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

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

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

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

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

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