If (\binom{n}{3}=\binom{n}{5}), what will be the value of (n)?

Correct answer: \(8\)

Use the symmetry property of combinations: \(\binom{n}{r}=\binom{n}{n-r}\). Here \(\binom{n}{3}=\binom{n}{5}\) and \(3\ne5\), so \(5=n-3\). Therefore, \(n=8\). For example, for \(n=7\), \(\binom{7}{3}=35\) but \(\binom{7}{5}=21\), so it is not correct. Exam tip: when \(\binom{n}{r}\) and \(\binom{n}{s}\) are equal with \(r\ne s\), first check whether \(r+s=n\).