First slide

Combinations

Question

If ‘n’ is an integer between 0 and 21, then the minimum value of n! (21 – n)! is

Moderate

A

9! 2!
B

10! 11!
C

20!
D

21!

Solution

$n! (21 - n)! = 21! \frac{n! (21 - n)!}{21!} = 21! \frac{1}{^{21} C_{n}} = \frac{21!}{^{21} C_{n}}$
For minimum value, $^{21} C_{n}$ is maximum.
Maximum value of $^{21} C_{n} =^{21} C_{21 - 1} =^{21} C_{10}$
$∴ Minimum value = \frac{21!}{^{21} C_{10}} = \frac{21!}{21!} \times 10! 11! = 10! 11!$

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