Final answer:
To evaluate the expression 8!6!/7!4!, common factorial terms are canceled out, resulting in a simplified expression of 8 × 30, which equals 240.
Step-by-step explanation:
To evaluate the expression 8!6!/7!4!, we need to simplify the factorials involved. To do so, we identify that a factorial such as n! (e.g., 7!) is a product of all positive integers from 1 up to n. By realizing that factorials can be broken down into their components, we can cancel out common terms.
Let's first simplify 8! and 7!:
- 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
- 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
Now, by dividing 8! by 7!, the numbers 7 through 1 will cancel out, leaving us with just 8.
Next, we simplify 6! and 4!:
- 6! = 6 × 5 × 4 × 3 × 2 × 1
- 4! = 4 × 3 × 2 × 1
By applying the same logic, we can reduce 6! by 4!, which cancels out the numbers 4 through 1, leaving us with 6 × 5 or 30.
The original expression now simplifies to:
8 × 30 = 240
Therefore, 8!6!/7!4! simplifies to 240, which is a reasonable result.