What is the value of (\binom{12}{5}-\binom{11}{5})?

Correct answer: 330

Using Pascal’s identity, \(\binom{n}{r}=\binom{n-1}{r}+\binom{n-1}{r-1}\), we get \(\binom{12}{5}=\binom{11}{5}+\binom{11}{4}\). Hence, \(\binom{12}{5}-\binom{11}{5}=\binom{11}{4}=330\). The value 462 is \(\binom{11}{5}\), not the required difference. Exam tip: for differences of adjacent binomial coefficients, apply Pascal’s identity to identify the remaining term directly.