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$\begingroup$ The theorem that $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ already assumes $0!$ is defined to be $1$. Otherwise this would be restricted to $0
Learn the definition, permutations, and formulas of zero factorial, which is the number of ways to arrange an empty set. Find out why zero factorial is equal to one and how it relates to other mathematical concepts.
Learn about the mathematical expression 00 and its different values in various contexts. Find out how 00 is defined in combinatorics, algebra, analysis, complex exponents, and computer programming.
Denise, You are correct that 0! = 1 for reasons that are similar to why x^0 = 1. Both are defined that way. But there are reasons for these definitions; they are not arbitrary. You cannot reason that x^0 = 1 by thinking of the meaning of powers as "repeated multiplications" because you cannot multiply x zero times.
Why is 0 factorial equal to 1 ProofBeginning with the definition of factorials we can work our way to a proof where 0! = 1 is mathematically proven.In the fi...
Learn what zero-factorial (0!) means and how it is defined as 1. Explore different proofs and examples of factorials and permutations using numbers and sets.
Learn the mathematical reason behind the rule that any number raised to the power of 0 equals 1. It's based on the rules of exponents and dividing by the base until only 1 remains.
The friend did not prove \(a^0=1\) from his definition, but from an unstated assertion that anything that preserves the properties is correct. In fact, if his definition had been, say, that exponentiation is defined by two axioms, that \(a^1=a\) and that \(a^m\cdot a^n=a^{m+n}\), that would be sufficient. See below for more.
However, when n = 0, the definition might seem problematic: 0! = 0 × (0 - 1) × (0 - 2) × … × 2 × 1. The issue arises because the product starts with 0, and it might appear that the result is 0. However, to maintain consistency and mathematical convenience, mathematicians define 0! to be 1. Now, let's prove this convention ...
For the equation to be true, we must force the value of zero factorial to equal 1, and no other. Otherwise, 1!≠1 which is a contradiction. So yes, 0! = 1 is correct because mathematicians agreed to define it that way (nothing more and nothing less) in order to be consistent with the rest of mathematics.