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why is zero factorial equal to one
Many people guess zero factorial is zero: no things, no arrangements. That's the myth. Factorial counts line-ups, and three things can line up in six ways.
Shrink the line: two things give two orders, one thing gives one. With zero things there is exactly one way to line them up: do nothing at all.
The pattern agrees. Each factorial is the next one divided by its own number: four factorial is twenty-four, divide by four for six, then two, then one, then one again.
And choosing needs it. Picking all three of three friends can be done in just one way, but the choosing formula gives one over zero factorial. So zero factorial must be one.
So it is a definition, but a forced one: three different roads land on one, and even the smooth curve through the factorials passes through one at zero.
Zero factorial is one because there is exactly one way to arrange nothing, and every factorial rule insists on it.