If `x + y = 1`, prove that `underset(r=0)overset(n)sum r.^(n)C_(r) x^(r ) y^(n-r) = nx`.

Options

A. n
B. np
C. npq
D. none of these

Answer

b

Explanation

(i) We have,
`sum_(r=1)^(n) r. ""^(n)C_(r) p^(r)q^(n-r)`
` sum_(r=1)^(n)r.(n)/(r)""^(n-1)C_(r-1) p.p^(r-1)q^((n-1)-(r -1))`
` = np {sum_(x=1)^(n) ""^(n-1)C_(r-1) p^(r - 1) q^((n-1)-(r-1))}`
` = np (q + p)^(n-1) [ because (q + p)^(n) = sum _(r=0)^(n) ""^(n) C_(r)p^(r) q^(n-r)]`
`= np [ because p + q = 1]` .