Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the factorial expression. $$\frac{12 !}{4 ! \cdot 8 !}$$

Step-by-Step Solution

Answer

The simplified expression of \( \frac{12 !}{8 ! \cdot 4 !} \) is 1

Step 1: Understanding the Factorial

The factorial, denoted as n!, is a mathematical expression that equals to the product of all positive integers less than or equal to n. More formally, \(n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1\). Now let's apply this to simplify the given expression.

Step 2: Expanding Factorials

First, we expand each factorial in the given expression separately. We have \(12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), \(4! = 4 \times 3 \times 2 \times 1\) and \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\). We notice that \(8! \times 4!\) equals to \(12!\) so the whole expression simplifies to 1.

Step 3: Calculate

Now we perform the calculation. The expression simplifies to \(\frac{12 !}{8 ! \cdot 4 !} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 4 \times 3 \times 2 \times 1} = 1\) given that all elements above and below the bar cancel each other out.