Questions: Evaluate. 7!2! / 8!4!

Solution Steps

To solve the given factorial expression, we need to simplify the numerator and the denominator separately and then divide them. Factorials can be simplified by canceling out common terms in the numerator and the denominator.

1. Write out the factorials in the numerator and the denominator.
2. Cancel out the common terms.
3. Simplify the remaining terms to get the final answer.

We start by writing out the factorials in the numerator and the denominator: \[ \frac{7! \cdot 2!}{8! \cdot 4!} \]

Next, we simplify the factorials by canceling out common terms. Note that \(8! = 8 \cdot 7!\), so we can cancel \(7!\) from both the numerator and the denominator: \[ \frac{7! \cdot 2!}{8 \cdot 7! \cdot 4!} = \frac{2!}{8 \cdot 4!} \]

Now, we calculate the remaining factorials: \[ 2! = 2 \quad \text{and} \quad 4! = 24 \]

Substitute the values back into the expression: \[ \frac{2}{8 \cdot 24} = \frac{2}{192} = \frac{1}{96} \]

Finally, we convert the fraction to its decimal form: \[ \frac{1}{96} \approx 0.01042 \]

Final Answer