If ` (1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)` ,
find the values of the following
`underset(0leile jlen)(sumsum)(i +j)(C_(i)pmC_(j) )^(2)`

Explanation

`therefore underset (i ne j)(sumsum)(C_(i)pm C_(j))^(2) = underset (0le iltjle n)(sumsum)(C_(i)^(2)C_(j)^(2)pm2C_(1)C_(j))`
`= underset (0leiltjlen)(sumsum)(C_(i)^(2) +C_(j)^(2))pm2 underset(0leiltjlen)(sumsum)C_(i) C_(j)`
`because underset(0leiltjlen)(sumsum)(C_(i) ^(2) + C_(j)^(2))`
`=( underset (0leiltjlen)overset(n " "n)(sumsum)(C_(i)^(2) +C_(j)^(2))-2sum_(i=0)^(n) C_(1)^(2))/(2) `
`(underset(i=0)overset(n)(sum)(underset(j=0)overset(n)(sum)C_(1)^(2) + underset(j=0)overset(n)(sum)C_(1)^(2)) - 2 *""^(2n)C_(n))/(2)`
`=( underset (i= 0)overset(n )(sum)((n+1)C_(1)^(2) + ""^(2n)C_(n))-2*""^(2n)C_(n))/(2)`
` ((n+1)sum_(i=0)^(n) C_(i)^(2) + ""^(2n)C_(n) sum_(i=0)^(n) 1-2*""^(2n)C_(n))/(2)`
` ((n+1)* ""^(2n)C_(n) +""^(2n)C_(n) *(n-1)-2.""^(2n)C_(n))/(2)`
` = n*""^(2n)C_(n)`
`therefore underset(oleiltjlen)(sumsum)(C_(i) pmC_(j))^(2) + n.""^(2n)C_(n) pm (2^(2n) - ""^(2n)C_(n)) " "` [from corollary 1]
`= (npm1) ""^(2n)C_(n) pm2^(2n) , underset(0leiltjlen)(sumsum) (i + j) C_(i) C_(j)`