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If x+y=1, prove that sum(r=0)^n r* ^nCr ...

If `x+y=1,` prove that `sum_(r=0)^n r* ^nC_r x^r y^(n-r)=nxdot`

Text Solution

Verified by Experts

We have
`underset(r=0)overset(n)sumr.^(n)C_(r)x^(r )y^(n-r) = underset(r=1)overset(n)sumn.^(n-1)C_(r-1)x^(r-1)x^(1)y^(n-r)`
` = nx underset(r=1)overset(n)sum .^(n-1)C_(r-1)x^(r-1)y^((n-1)-(r-1))`
`= nx(x+y)^(n-1)`
` = nx , [:' x + y = 1]`
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