Problem 25 - College Algebra

Question

Evaluate each factorial expression. $$\frac{16 !}{2 ! 14 !}$$

Step-by-Step Solution

Answer

The value of the expression \(\frac{16 !}{2 ! 14 !}\) is 120.

Step 1: Recognize the expression

Notice that the expression is in the form of the combination formula. Here, \(n=16\) and \(k=2\). It can be rewritten as \(C(16, 2)\).

Step 2: Use the formula for combinations

Apply the combination formula, which states that the number of combinations of n items taken k at a time \(C(n, k)\) is equal to the number of combinations of n items taken n-k at a time, that is, \(C(n, k) = C(n, n-k)\). Here, \(C(16, 2) = C(16, 16 - 2) = C(16,14)\). The number of ways 16 items can be picked 14 at a time is the same as the number of ways those items can be picked 2 at a time.

Step 3: Calculate the combination

Use the combination formula to do the calculation. \(C(n, k) = \frac{n!}{k!(n-k)!}\), which simplifies to \(C(16,2) = \frac{16!}{2!(16-2)!} = \frac{16!}{2!14!}\). This simplifies further to \(C(16,2) = \frac{16*15}{2*1} = 120\).