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nCr Definitions and Properties

If nPr = nP...

Question

If $^{n} P_{r} =^{n} P_{r}_{+}_{1}$ and $^{n} C_{r} =^{n} C_{r}_{-}_{1}$ , then the values of n and r are:

A

r, 3

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B

3, 2

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C

4, 2

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D

3, 4

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Solution

The correct option is B $3, 2$
$^{n} P_{r} =^{n} P_{r + 1}$
$\frac{n!}{(n - r)!} = \frac{n!}{(n - (r + 1))!}$
$(n - r - 1)! = (n - r)!$
$n - r = 1$ -------------(1)
$^{n} C_{r} =^{n} C_{r - 1}$
$\frac{n!}{(n - r)! r!} = \frac{n!}{(n - r + 1)! (r + 1)!}$
$(r - 1)! (n + 1 - r)! = (n - r)! r!$
$n + 1 - r = r$
$n + 1 = 2 r$
$n = 2 r - 1$ -------------(2)
By solving both equations, we get
$r = 2$
$\Rightarrow n = 2 (2) - 1 = 3$

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