What is the value of (\frac{7!}{5!,2!})?

Correct answer: (21)

The direct answer is B, 21. A factorial means multiplying all positive integers down to 1: \(7!=7\times6\times5\times4\times3\times2\times1\). In the expression, cancel \(5!\) from \(7!\): \(\frac{7!}{5!\,2!}=\frac{7\times6\times5!}{5!\times2}=\frac{7\times6}{2}=\frac{42}{2}=21\). Option A, 14, is wrong because it does not result from the required factorial cancellation. Option B, 21, is correct because it exactly equals the simplified expression. Option C, 35, is wrong; it may come from incomplete or incorrect multiplication and division. Option D, 42, is also wrong because 42 is the numerator \(7\times6\) before dividing by \(2!=2\). Remember: cancel the smaller factorial first, then divide by the remaining factorial.