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If (1+x)^n=C0+C1x+C2x^2++Cn x^n , then s...

If `(1+x)^n=C_0+C_1x+C_2x^2++C_n x^n` , then show that the sum of the products of the coefficients taken two at a time, represented by `sumsum_(0lt=i

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We know that ,
`2sum _(0) sum_(leile j lt n)sum C_i C_j=sum_(i=0)^(n)sum_(j=0)^(n)C_(i=0)^(n)sum_(j=0)^(n)C_i C_J- sum_(i=0)^(n)sum_(j=0)^(n)C_i C_j =sum_(i=0)^(n)C_i sum_(j=0)^(n)C_(j) -sum_(i=0)^(n)C_(i)^(2)`
`=2^n 2^n -(.^2n C_n)=2^2n -.^2n C_n`
`therefore sum sum_(0 le i n j le n) C_iC_j=(2^2n -.^2n C_N)/(2)=2^(2n-1)-((2n)!)/(2(n!))`.
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