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Evaluate sum(i=0)^(n)sum(j=0)^(n) ""^(n)...

Evaluate `sum_(i=0)^(n)sum_(j=0)^(n) ""^(n)C_(j) *""^(j)C_(i), ile j ` .

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We have , `sum_(i=0)^(n)sum_(j=0)^(n) ""^(n)C_(j) *""^(j)C_(i)`
`= ""^(n)C_(1) (""^(1)C_(0) + ""^(1)C_(1)) +""^(n)C_(2) (""^(2)C_(0) + ""^(2)C_(1) + ""^(2)C_(2))`
` + ""^(n)C_(3) (""^(3)C_(0) + ""^(3)C_(1) + ""^(3)C_(2) + ""^(3)C_(3))`
` + ""^(n)C_(4) (""^(4)C_(0) + ""^(4)C_(1) + ""^(4)C_(2) + ""^(4)C_(3)+""^(4)C_(4))`
`+ ...+ ""^(n)C_(n)(""^(n)C_(0) + ""^(n)C_(1) + ""^(n)C_(2) + ...+ ""^(n)C_(n))`
`=""^(n)C_(1)(2)+""^(n)C_(2)(2)^(2) + ""^(n)C_(3)(2)^(3) + ... + ""^(n)C_(n)(2)^(n) = (1+ 2)^(n) - 1 `
`= 3^(n) -1`
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ARIHANT MATHS-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
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  5. For natural numbers m ,n ,if(1-y)^m(1+y)^n=1+a1y+a2y^2+ ,a n da1=a2=10...

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  6. In the binomial expansion of (a - b)^(n) , n ge 5 , the sum of the ...

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  7. The sum of the series C 20()0-C 20()1+C 20()2-C 20()3+...-...+C 20()(1...

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  8. Statement-1: sum(r =0)^(n) (r +1)""^(n)C(r) = (n +2) 2^(n-1) Stat...

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  9. The remainder left out when 8^(2n)""(62)^(2n+1) is divided by 9 is (1)...

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  10. For r= 0, 1,.....,10, let Ar,Br, and Crdenote, respectively, the coe...

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  11. So, statement-1 is also true. Stetement-2 is a correct expanation fo...

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  12. The coefficient of x^(7) in the expansion of (1-x-x^(2) + x^(3))^(6) i...

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  13. If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is ...

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  14. In the expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x-x^(1/2)))^10 ...

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  15. The coefficients of three consecutive terms of (1+x)^(n+5) are in the ...

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  16. If the coefficients of x^(3) and x^(4) in the expansion of (1 + ax + b...

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  17. Coefficient of x^(11) in the expansion of (1+x^2)(1+x^3)^7(1+x^4)^(12)...

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  18. The sum of coefficients in integral powers of x in the binominal expan...

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  19. The coefficient of x^9 in the expansion of (1+x)(16 x^2)(1+x^3)(1+x^(1...

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  20. If the number of terms in the expansion of (1-2/x+4/(x^2))^n , x!=0, i...

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  21. Let m be the smallest positive integer such that the coefficient of x^...

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