$(x + a)^{n} = x^{n} +^{n} C_{1} x^{n - 1} a +^{n} C_{2} x^{n - 2} a^{2} + . . . . . . . . . . +^{n} C_{n} a^{n}$
Then coefficient of $x^{n - r} a^{r}$ .

\frac{n (n - 1) . . . . . . (n - r + 1)}{r!}

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\frac{n (n - 1) . . . . . . (n - r)}{r!}

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\frac{n!}{(n - r)! r!}

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\frac{n (n - 1) (n - 2) . . . . . . (n - r)}{(n - r)! r!}

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Solution

The correct options are
A $\frac{n (n - 1) . . . . . . (n - r + 1)}{r!}$
C $\frac{n!}{(n - r)! r!}$
The coefficient of $x^{n - r} a^{r}$ is $^{n} C_{r}$ .
$^{n} C_{r}$ $=$ $\frac{n!}{(n - r)! r!}$
$^{n} C_{r}$ $=$ $\frac{n (n - 1) (n - 2) . . (n - r + 1) (n - r)!}{(n - r)! r!}$
$^{n} C_{r}$ $=$ $\frac{n (n - 1) . . . . (n - r + 1)}{r!}$
Thus both option A and c is correct.

Suggest Corrections

Similar questions

Q. Prove that:
(i)

\frac{n!}{(n - r)!}

= n (n − 1) (n − 2) ... (n − (r − 1))

(ii)

\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}

Prove that:
(i) $\frac{n!}{(n - r)!}$
= n(n-1)(n-2)...(n-(r-1))
(ii) $\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!}$
= $\frac{(n + 1)!}{r! (n - r + 1)!}$

Q. Prove that:

\frac{n!}{(n - r)!} = n (n - 1) (n - 2) . . . (n - (r - 1))

Q. If

(1 - x)^{- n} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . . . . + a_{r} x^{r} + . . . . .

, then

a_{0} + a_{1} + a_{2} + . . . . . . + a_{r}

is equal to

$r^{t h}$ term in the expansion of $(a + 2 x)^{n}$ is

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(x+a)n = xn+ nC1xn−1a+ nC2xn−2a2+..........+nCnanThen coefficient of xn−rar.

The correct options are A n(n−1)......(n−r+1)r! C n!(n−r)!r!The coefficient of xn−rar is nCr. nCr=n!(n−r)!r!nCr =n(n−1)(n−2)..(n−r+1)(n−r)!(n−r)!r!nCr=n(n−1)....(n−r+1)r!Thus both option A and c is correct.

$(x + a)^{n} = x^{n} +^{n} C_{1} x^{n - 1} a +^{n} C_{2} x^{n - 2} a^{2} + . . . . . . . . . . +^{n} C_{n} a^{n}$
Then coefficient of $x^{n - r} a^{r}$ .