The Correct Answer and Explanation is:
The correct answer is D. Rewrite 900! as 900 * (899!) and then cancel out 899! from the numerator and denominator.
To evaluate the expression 900!/899! without a calculator, the most efficient method relies on the definition of a factorial. A factorial, represented by the symbol “!”, is the product of all positive integers up to that number. For example, 5! equals 5 × 4 × 3 × 2 × 1.
The expression we need to solve is the division of 900! by 899!. Let’s look at what each factorial represents:
- 899! = 899 × 898 × 897 × … × 3 × 2 × 1
- 900! = 900 × 899 × 898 × 897 × … × 3 × 2 × 1
The key insight is to notice that the expansion of 900! contains the entire expansion of 899!. Therefore, we can rewrite the larger factorial, 900!, in terms of the smaller one, 899!.
We can express 900! as:
900! = 900 × (899 × 898 × 897 × … × 1)
Since the expression in the parentheses is the definition of 899!, we can substitute it:
900! = 900 × (899!)
Now, we can replace 900! in the original fraction with this new expression:
Expression = (900 × 899!) / 899!
We now have a common factor of 899! in both the numerator and the denominator. This common factor can be cancelled out, simplifying the expression significantly.
After cancellation, we are left with:
Expression = 900
This method avoids the impossible task of calculating the actual values of 900! and 899! by hand and provides a simple, direct answer. The other options are mathematically incorrect. For instance, you cannot subtract a term from the numerator and denominator (Option A) or simply cancel the factorial symbol (Option C).
