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Find the sum underset(0leiltjlen)(sumsum...

Find the sum `underset(0leiltjlen)(sumsum).^(n)C_(i)`.

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`underset(0leiltjlen)(sumsum)j..^(n)C_(i)`
`= underset(r=0)overset(n-1)sum.^(n)C_(r)[(r+1)+(r+2)+"...."+(n)]`
`= underset(r=0)overset(n-1)sum.^(n)C_(r)[(r+1)+(r+2)+"...."(r+(n-r))]`
`= underset(r=0)overset(n-1)sum.^(n)C_(r)(n-r)/(2)(r+1+n)`
`= 1/2 underset(r=0)overset(n)sum.^(n)C_(r) (n(n+1)-r-r^(2))`
` = 1/2 [n(n+1)underset(r=0)overset(n)sum.^(n)C_(r)-underset(r=0)overset(n)sumr^(n)C_(r)-underset(r=0)overset(n)sumr^(2)..^(n)C_(r)]`
`=1/2[n(n+1).2^(n)-n underset(r=0)overset(n)sum.^(n-1)C_(r-1)-n underset(r=0)overset(n)sumr..^(n-1)C_(r-1)]`
`=1/2[n(n+1).2^(n)-n.2^(n-1)-n underset(r=0)overset(n)sum((n-1)..^(n-2)C_(r-2)+.^(n+1)C_(r-1))]`
`= 1/2[n(2n+1).2^(n-1)-n(n-1).2^(n-2)-n.2^(n-1)]`
`= n(3n+1).2^(n-3)`
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