Prove that-i n -n-r -nn-1n-2 -n-r-1ii n -n-r - r -n -n-r-1 -r-1 -11-172- r - n - r -1

(i) We have, LHS = n!/(n r)! = n(n 1)(n 2)(n 3)...(n r+2)(n r+1)(n r)!/(n r!) = n(n 1)(n 2)(n 3)...(n r+2)(n r+1) = n(n 1)(n 2)(n 3)...((n (r 2)) (n (r 1)) = n ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Permutation: n Different Things Taken r at a Time

Prove that:i ...

Question

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

Open in App

Solution

(i) We have,
LHS = $\frac{n!}{(n - r)!}$
= $\frac{n (n - 1) (n - 2) (n - 3) . . . (n - r + 2) (n - r + 1) (n - r)!}{(n - r)!}$
= $n (n - 1) (n - 2) (n - 3) . . . (n - r + 2) (n - r + 1)$
= $n (n - 1) (n - 2) (n - 3) . . . ((n - (r - 2)) (n - (r - 1))$
= $n (n - 1) (n - 2) (n - 3) . . . (n - (r - 1))$
= RHS
$∴$ LHS =RHS
(ii) To prove $\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!} = \frac{(n + 1)!}{r! (n - r + 1)!}$
LHS= $\frac{n!}{(n - r)! r!} + \frac{n!}{(n - r + 1)! (r - 1)!}$
= $\frac{n!}{(n - r)! (r) [(r - 1)]!} + \frac{n!}{(n - r + 1) [(n - r)!] (r - 1)!}$
= $\frac{n!}{(n - r)! \times (r - 1)!} [\frac{1}{r} + \frac{1}{n - r + 1}]$ = $\frac{n!}{(n - r)! \times (r - 1)!} [\frac{n - r + 1 + r}{r (n - r + 1)}]$
= $\frac{n!}{(n - r)! \times (r - 1)!} [\frac{n + 1}{r (n - r + 1)}]$ = $\frac{(n + 1) \times n!}{(n - r + 1) \times (n - r)! \times r \times (r - 1)!}$
= $\frac{(n + 1)!}{(n - r + 1)! \times r!} = \frac{(n + 1)!}{r! (n - r + 1)!}$
=RHS
$∴$ LHS = RHS

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)!}

Q. Prove that:

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

Prove that:

n! / r! x (n-r)! + n! / (r-1)! x (n-r+1) = (n+1)! / r! x (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

Q. Prove that

^{n}

_{r}

^{n - 1}

_{r}

^{n - 2}

_{r}

^{n - 2}

_{r}

+....................+

^{r}

_{r}

^{n + 1}

_{r + 1}

Join BYJU'S Learning Program

Prove that: (i) n!(n−r)! = n(n-1)(n-2)...(n-(r-1)) (ii) n!(n−r)!r!+n!(n−r+1)!(r−1)! = (n+1)!r!(n−r+1)!

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