Prove that $^{n} C_{r} =^{n} C_{n - r}$ .

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Solution

$^{n} C_{r} =^{n} C_{n - r}$ .
The number of combinations of n dissimilar things taken r at a time will be $^{n} C_{r}$ . Now if we take out a group of r things, we are left with a group of (n-r) things. Hence the number of combinations of n things taken r at a time is equal to the number of combinations of n things taken (n-r) at a time.
$∴$ $^{n} C_{r} =^{n} C_{n - r}$ .
Alternative method :
$^{n} C_{r} = \frac{n!}{r! (n - r)!}$
$^{n} C_{n - r} = \frac{n!}{(n - r)! (n - n + r)!} = \frac{n!}{r! (n - r)!}$
$∴$ $^{n} C_{r} =^{n} C_{n - r}$ .
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