What is the value of (7!/5!) × (5!/6!)?

Correct answer: 7

The governing concept is the factorial identity k! = k(k−1)! and cancellation of common nonzero factors. In (7!/5!)×(5!/6!), the 5! in the numerator of the second fraction cancels the 5! in the denominator of the first fraction. The expression therefore becomes 7!/6!. Using 7!=7×6!, we obtain 7!/6!=7. A separate check gives 7!/5!=7×6=42 and 5!/6!=1/6, so their product is 42/6=7. Hence option C is correct. The value 6 results from retaining only one factor from the first ratio, while 8 would reflect an incorrect factorial step. Careful cancellation preserves the multiplication structure and confirms that the answer is uniquely 7.