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The value of ^n C1+^(n+1)C2+^(n+2)C3++^(...

The value of `^n C_1+^(n+1)C_2+^(n+2)C_3++^(n+m-1)C_m` is equal to `^m+n C_(n-1)` `^m+n C_(n-1)` `^mC_(1)+^(m+1)C_2+^(m+2)C_3++^(m+n-1)` `^m+1C_(m-1)`

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The value of ^n C_1+^(n+1)C_2+^(n+2)C_3++^(n+m-1)C_m is equal to (a)^m+n C_(n-1) (b)^m+n C_(n-1) (c)^mC_(1)+^(m+1)C_2+^(m+2)C_3++^(m+n-1) (d)^m+1C_(m-1)

Prove that mC_(1)^(n)C_(m)-^(m)C_(2)^(2n)C_(m)+^(m)C_(3)^(3n)C_(m)-...=(-1)^(m-1)n^(m)

^(n)C_(m)+^(n-1)C_(m)+^(n-2)C_(m)+............+^(m)C_(m)

The value of the determinant |(1,1,1),(.^(m)C_(1),.^(m +1)C_(1),.^(m+2)C_(1)),(.^(m)C_(2),.^(m +1)C_(2),.^(m+2)C_(2))| is equal to

C_(0)-3C_(1)+5c_(3)+....+(-1)^(n)(2n+1)C_(n) is equal to

Using binomial theorem (without using the formula for sim nC_(r)), prove that ^nC_(4)+^(m)C_(2)-^(m)C_(1)^(n)C_(2)=^(m)C_(4)-^(m+n)C_(1)^(m)C_(3)+^(m+n)C_(2)^(m)C_(2)-^(m+n)C_(3)^(m)C_(1)+^(m+n)C_(4)

Find the value of sum_(p=1)^(n)(sum_(m=p)^(n)C_(m)^(m)C_(p)) And hence,find the value of lim_(n rarr oo)(1)/(3^(n))sum_(p=1)^(n)(sum_(m=p)^(n)C_(m)^(m)C_(p))

A DAS GUPTA-Binomial Theorem for Positive Integrel Index-Exercise
  1. (b) Find the value of sum(r=m)^n .^rCm,n>m

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  2. Evaluate (""^3C3+""^4C3+""^5C3+...+""^nC3)xx(""^nC3+""^nC4+""^nC5+...+...

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  3. The value of ^n C1+^(n+1)C2+^(n+2)C3++^(n+m-1)Cm is equal to ^m+n C(n-...

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  4. Prove that ""^(n+1)C2+2*sum(k=2)^n""^kC2=sum(k=1)^nk^2

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  5. If (1+x)^n=C0+C1x+C2x^2+...+Cnx^n , find the sum of the following seri...

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  6. Prove that ""^nC0+2*""^nC1+3*""^nC2+...+(n+1)""^nCn=(n+2)2^(n-1)

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  7. Prove that ""^nC0+3*""^nC1+5*""^nC2+...+(2n+1)""^nCn=(n+1)2^(n)

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  8. Prove that ""^nC0-2*""^nC1+3*""^nC2-...+(-1)""^n(n+1)""^nCn=0

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  9. If sn=""^nC0+2*""^nC1+3*""^nC2+...+(n+1)*""^nCn then find sum(n=1)^oos...

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  10. Find the sum :1*""^nC0+2*""^nC1+3*""^nC2+4*""^nC3+…, where n is an odd...

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  11. Show that :""^nC0*m-""^nC1*(m-1)+""^nC2*(m-2)-...+(-1)^n*""^nCn*(m-n)=...

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  12. Evaluate sum(r=1)^npr/r*"^nCr where pr denotes the sum of the first r ...

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  13. Prove by binomial expansion that sum(k=1)^nk^2*"^nCk=n(n+1)2^(n-2)

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  14. Evaluate sum(r=0)^n(r+1)^2*"^nCr

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

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

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

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

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

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