What is the value of \( \frac{24!}{21!\cdot3!}-\frac{23!}{20!\cdot3!} \)?

Correct answer: 253

The first term is \(\frac{24!}{21!\cdot3!}=\binom{24}{3}\), and the second term is \(\frac{23!}{20!\cdot3!}=\binom{23}{3}\). By Pascal’s identity, \(\binom{n}{r}-\binom{n-1}{r}=\binom{n-1}{r-1}\), the expression becomes \(\binom{23}{2}\). Thus, \(\binom{23}{2}=\frac{23\times22}{2}=253\), so option C is correct. The value 252 may result from an incorrect calculation after applying the identity. Exam tip: in differences of consecutive binomial coefficients, look for Pascal’s identity first.