The least positive integer n for which n−1C5+n−1C6

The least positive integer $n$ for which $^{n - 1} C_{5} +^{n - 1} C_{6} <^{n} C_{7}$ is

A
14
B
15
C
16
D
28

Solution:

Since $^{m} C_{r - 1} +^{m} C_{r} =^{m + 1} C_{r}$ we can write the given inequality as

$^{n} C_{6} <^{n} C_{7}$

$\Rightarrow \frac{^{n} C_{6}}{^{n} C_{7}} < 1 \Rightarrow \frac{7}{n - 6} < 1 \Rightarrow n > 13$

$∴$ the value of $n$ is 14.

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