Evaluate 1000!/998! without a calculator.

Verified step by step guidance

1. Recall the definition of factorial: for any positive integer \(n\), \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
2. Express both factorials explicitly: \(1000! = 1000 \times 999 \times 998!\) and \$998!$ remains as is.
3. Rewrite the expression \(\frac{1000!}{998!}\) by substituting the expanded form of \$1000!$:
4. \[\frac{1000!}{998!} = \frac{1000 \times 999 \times 998!}{998!}\]
5. Cancel out the common \$998!$ terms in numerator and denominator, leaving:
6. \[1000 \times 999\]

Key Concepts

Factorial Definition

The factorial of a positive integer n, denoted n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Understanding this definition is essential to simplify expressions involving factorials.

Factorial Simplification Using Division

When dividing factorials like n! / (n - k)!, many terms cancel out because (n - k)! is a part of n!. This allows simplification to a product of k consecutive integers starting from n downwards, making calculations easier without full expansion.

Properties of Factorials in Ratios

Ratios of factorials often reduce to products of a few terms. For example, 1000! / 998! equals 1000 × 999 because 998! cancels out the terms from 1 to 998 in the numerator. Recognizing this property helps evaluate large factorial expressions efficiently.