If $^{n} C_{10} =^{n} C_{15}$ , find $^{27} C_{n}$ .

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Solution

$^{n} C_{10} =^{n} C_{15}$ . ....(1)
We know that $^{n} C_{r} =^{n} C_{n - r}$ ,
$^{n} C_{10} =^{n} C_{n - 10} =^{n} C_{15}$ , ....by (1)
$∴$ n - 10 =15 or n = 25
or If $^{n} C_{x} =^{n} C_{y}$ then x + y = n.
Hence from (1), we get 10 + 15 = 25= n
$∴$ $^{27} C_{n} =^{27} C_{25} =^{27} C_{27 - 25} =^{27} C_{2}$
= $\frac{27 \times 26}{1 \times 2} = 27 \times 13 = 351$

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