If $^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}$ denote the binomial coeffients in the expansion of $(1 + x)^{n}$ and $p + q = 1,$ then $n \sum r = 0 r^{n} C_{r} p^{r} q^{n - r}$ is

n p^{2}

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n p q

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n p

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None of these

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Solution

The correct option is C $n p$
we have,
$n \sum r = 0 r^{n} C_{r} p^{r} q^{n - r}$
$= n \sum r = 0 r . \frac{n}{r} .^{n - 1} C_{r - 1} . p . p^{r - 1} . q^{(n - 1) - (r - 1)}$
$= n p {n \sum r = 0^{n - 1} C_{r - 1} . p^{r - 1} . q^{(n - 1) - (r - 1)}}$
$= n p (q + p)^{n - 1}$
$= n p (∵ p + q = 1)$

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Similar questions

Q. If

^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}

denote the binomial coeffients in the expansion of

(1 + x)^{n}

and

p + q = 1,

then

n \sum r = 0 r^{n} C_{r} p^{r} q^{n - r}

Q. If

^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}

denote the binomial coefficients in the expansion of

(1 + x)^{n}

and

p + q = 1

, then

n \sum r = 0 r^{2}^{n} C_{r} p^{r} q^{n - r}

Find the coefficient of $x^{n - 2}$ in
$(^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n}) \times (^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} . . . . .^{n} C_{n} x^{n})$

Q. IF

S_{n} =^{n} C_{0} .^{n} C_{1} +^{n} C_{1} .^{n} C_{2} + . . . . +^{n} C_{n - 1} .^{n} C_{n} a n d \frac{S_{n + 1}}{S_{n}} = \frac{15}{4}

, then n =

Q. Let

A_{n} =^{n} C_{0}^{n} C_{1} +^{n} C_{1}^{n} C_{2} + . . . +^{n} C_{n - 1}^{n} C_{n}

and

\frac{A_{n + 1}}{A_{n}} = \frac{15}{4}

,then

n

equals

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If nC0, nC1, nC2,⋯, nCn denote the binomial coeffients in the expansion of (1+x)n and p+q=1, then n∑r=0r nCr prqn−r is

The correct option is C npwe have, n∑r=0r nCr prqn−r =n∑r=0r.nr. n−1Cr−1. p. pr−1.q(n−1)−(r−1) =np{n∑r=0 n−1Cr−1. pr−1.q(n−1)−(r−1)} =np(q+p)n−1 =np (∵p+q=1)

If $^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}$ denote the binomial coeffients in the expansion of $(1 + x)^{n}$ and $p + q = 1,$ then $n \sum r = 0 r^{n} C_{r} p^{r} q^{n - r}$ is

The correct option is C $n p$
we have,
$n \sum r = 0 r^{n} C_{r} p^{r} q^{n - r}$
$= n \sum r = 0 r . \frac{n}{r} .^{n - 1} C_{r - 1} . p . p^{r - 1} . q^{(n - 1) - (r - 1)}$
$= n p {n \sum r = 0^{n - 1} C_{r - 1} . p^{r - 1} . q^{(n - 1) - (r - 1)}}$
$= n p (q + p)^{n - 1}$
$= n p (∵ p + q = 1)$