💯 Maths

# Why in the rules of factorial 0! = 1?

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Sam Georgiou1st in integrated masters (MMath) at Durham University.15 students helped

I can think of five main reasons as to why 0! is defined to be 1. Though bear in mind this is just a definition, we could have defined it to be anything we want, or even leave it as defined but that would be silly because we want things to be neat and crisp and consistent, and it turns out defining it to be 1 does just that. The reasons: Reason 1: Suppose we have objects A,B,C then we have 3!=6 ways of arranging the objects {ABC,ACB,BCA,BAC,CAB,CBA} now remove object C, now there are two objects and 2!=2 ways to arrange them {AB, BA} Do the same again and we have one thing and 1!=1 ways of arranging it {A} Finally we do it one last time, and we define 0! to be "the number of ways we can arrange no objects" which is precisely 1. This abstract idea crops up a lot in maths, the idea that when there is nothing to do or nothing to check then it still means something (usually in the context of identity elements). But I'll admit this is a bit confusing, so reason 2. Reason 2: 2!=3!/3=6/3=2 1!=2!/2=2/2=1 0!=1!/1=1/1=1 That motivates the definition. Reason 3: The gamma function. There is an integral function which generalises the idea of the factorial so that the function agrees with the factorial function on every positive integer, but allows us to extend the factorial function to all real (fractions and negatives and irrationals) numbers and Gamma(0)=1, so we set 0! to 1 as well Reason 4: Binomial coefficients, read up on how these are define