Why is 5 factorial equal to 120, and does it matter?

It is not a coincidence. $5!$ means the number of ways to arrange 5 distinct objects. There are: $$5$$ choices for the first position, then: $$4$$ choices for the second, then $3$, then $2$, then $1$. So: $$5! = 5\cdot 4\cdot 3\cdot 2\cdot 1 = 120.$$ That number appears whenever order matters. For example, 5 different books can be placed on a shelf in 120 orders, and 5 runners can finish a race in 120 possible rankings. The deeper pattern is that factorials count permutations. They also appear in combinations: $$\binom{n}{k}=\frac{n!}{k!(n-k)!},$$ Taylor series, probability, and many counting problems. So $5!=120$ is not just arithmetic. It is the first place where you start to feel how quickly arrangements grow.