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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 ENGLISH-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
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  7. The sum of series ^^(20)C0-^^(20)C1+^^(20)C2-^^(20)C3++^^(20)C 10 is 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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  11. Let S(1) = sum(j=1)^(10) j(j-1).""^(10)C(j), S(2) = sum(j=1)^(10)j."...

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

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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. The term independent of x in expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(...

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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 coefficient of x^(3) and x^(4) in the expansion of (1+ax+bx^(2)...

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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 coefficient of integral powers of x in the binomial expansi...

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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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