The reason 0! equals 1 comes down to a combination of mathematical consistency, combinatorics, and the definition of the factorial function itself. It ensures that many formulas and identities involving factorials work correctly.
Maintaining Consistency
The factorial function, denoted by n!, is typically defined for non-negative integers as the product of all positive integers less than or equal to n. This means:
n! = n (n-1) (n-2) ... 2 * 1
We can also define it recursively as:
- n! = n * (n-1)!
If we apply this definition to 1!, we get:
1! = 1 (1-1)! = 1 0!
If we want this recursive definition to hold true, and we know that 1! = 1, then:
1 = 1 * 0!
Therefore, 0! must equal 1.
Combinatorial Interpretation
From a combinatorial perspective, n! represents the number of ways to arrange n distinct objects. For example, 3! = 6 because there are 6 ways to arrange 3 distinct objects (ABC, ACB, BAC, BCA, CAB, CBA).
What does it mean to arrange 0 objects? There is only one way to arrange nothing - you simply don't arrange anything! This single possibility corresponds to 0! = 1. This can be thought of as the number of ways to arrange an empty set, which is 1.
Practical Applications
Defining 0! = 1 is crucial for various mathematical formulas and identities, particularly in areas like:
- Combinations: The formula for combinations, nCr = n! / (r! * (n-r)!), would be undefined when r = n if 0! was not equal to 1.
- Taylor Series: The Taylor series expansion of many functions relies on factorial terms. Defining 0! = 1 allows these series to be valid at x = 0.
In essence, setting 0! to 1 doesn't arise arbitrarily; it's a necessary definition to maintain mathematical coherence and consistency within the broader framework of combinatorics and other mathematical fields.
