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

Prove that `(""^nC_1)/(""^nC_0)+2*(""^nC_2)/(""^nC_1)+3*(""^nC_3)/(""^nC_2)+...+n*(""^nC_n)/(""^nC_(n-1))=frac{n(n+1)}{2}`

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Prove that ""^nC_0-2*""^nC_1+3*""^nC_2-...+(-1)""^n(n+1)""^nC_n=0

Prove that ""^nC_0+3*""^nC_1+5*""^nC_2+...+(2n+1)""^nC_n=(n+1)2^(n)

Prove that ""^nC_0+2*""^nC_1+3*""^nC_2+...+(n+1)""^nC_n=(n+2)2^(n-1)

Prove that 2*""^nC_0+2^2*(""^nC_1)/2+2^3*(""^nC_2)/3+...2^(n+1)*(""^nC_n)/(n+1)=(3^(n+1)-1)/(n+1)

Prove that : (1+""^nC_1+""^nC_2+""^nC_3+...+""^nC_n)^2=1+""^(2n)C_1+""^(2n)C_2+""^(2n)C_3+...+""^(2n)C_(2n)

Evaluate (""^3C_3+""^4C_3+""^5C_3+...+""^nC_3)xx(""^nC_3+""^nC_4+""^nC_5+...+""^nC_n)

Prove that (nC_(0))/(1)+(^nC_(2))/(3)+(^nC_(4))/(5)+(^nC_(6))/(7)+...=(2^(n))/(n+1)

Prove that "^nC_r + 2. ^nC_(r-1) + ^nC_(r-2) = ^(n+2)C_r

Prove that: (1)/(2)nC_(1)-(2)/(3)nC_(2)+(3)/(4)nC_(3)-(4)/(5)nC_(4) +....+((-1)^(n+1)n)/(n+1)*nC_(n)=(1)/(n+1)

Show that : ""^nC_0*m-""^nC_1*(m-1)+""^nC_2*(m-2)-...+(-1)^n*""^nC_n*(m-n)=0

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. 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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  2. Use a combinatorial argument to prove that (C(n,1))^2+2(C(n,2))^2+3(C...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  20. The number of terms in the expansion of (1+x^(1/5))^55 which are free ...

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