Simplify \(\frac{^{n}P_{4}}{^{n}C_{4}}\)

Answer Details

The expression \(\frac{^{n}P_{4}}{^{n}C_{4}}\) represents the number of ways to choose and arrange 4 objects out of a total of n objects, divided by the number of ways to choose 4 objects out of n objects without arranging them. The number of ways to choose and arrange 4 objects out of n objects is given by the formula for permutations, which is \(^{n}P_{4} = \frac{n!}{(n-4)!}\). The number of ways to choose 4 objects out of n objects without arranging them is given by the formula for combinations, which is \(^{n}C_{4} = \frac{n!}{4!(n-4)!}\). So, \(\frac{^{n}P_{4}}{^{n}C_{4}} = \frac{\frac{n!}{(n-4)!}}{\frac{n!}{4!(n-4)!}} = \frac{4!}{1!} = 24\). Therefore, the answer is 24.