Problem 25 - College Algebra

Question

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

Step-by-Step Solution

Answer

The evaluated factorial expression is 120.

Step 1: Simplify Factorials

Start by writing the expression with 16! in expanded form. Skip expanding 2! and 14! at first because these numbers can potentially simplify the larger factorial 16!, hence making calculations easier. The expression becomes: \(\frac{16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{2 × 1 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}\).

Step 2: Cancel Out Common Factors

You can cancel out the common factors in the numerator and denominator. Notice that the factors from 14 to 1 in the numerator are the same as those in the denominator. After cancelation, the expression simplifies to: \(\frac{16 × 15}{2 × 1}\).

Step 3: Perform Calculation

Now compute the values. \(\frac{16 × 15}{2 × 1} = \frac{240}{2} = 120\).