If (\binom{n}{5}=\binom{n}{7}), what will be the value of (n)?

Correct answer: 12

Use the symmetry property of combinations: \(\binom{n}{r}=\binom{n}{n-r}\). For \(\binom{n}{5}=\binom{n}{7}\), we need \(7=n-5\). Thus, \(n=5+7=12\). Hence, 12 is the correct option. For example, when \(n=11\), \(\binom{11}{5}\) and \(\binom{11}{7}=\binom{11}{4}\) are not equal. Exam tip: for equal binomial coefficients, check whether the lower indices are complementary, i.e. \(r\) and \(n-r\).