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Pascal Triangle

If n +1 C 5- ...

Question

If $^{n + 1} C_{5} -^{n} C_{4} >^{n} C_{3}$ , then the minimum value of $n$ is

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Solution

$^{n + 1} C_{5} -^{n} C_{4} >^{n} C_{3} \Rightarrow^{n + 1} C_{5} >^{n} C_{4} +^{n} C_{3} \Rightarrow^{n + 1} C_{5} >^{n + 1} C_{4} (∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}) \Rightarrow \frac{^{n + 1} C_{5}}{^{n + 1} C_{4}} > 1 \Rightarrow \frac{n - 3}{5} > 1 \Rightarrow n > 8$

Hence, the minimum value of $n$ is $9.$

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