# Simplifying a fraction?

$$\frac {n-2}{n} \cdot \frac {n-3}{n-1} \cdot \frac {n-4}{n-2} \cdots \frac{2}{4} \cdot \frac{1}{3} = \frac {1}{n(n-1)}$$

Why is this true? Notice the denominators and numerators cancel out, but since they "aren't in sync" the first two denominators and the last two numerators will not be cancelled out. Considering this, shouldn't the product be:

$$\frac{2}{n(n-1)}$$

What am I misunderstanding?

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Yes, it should be $$\dfrac2{n(n-1)}$$ To see this take $n = 4$, we then get that $$\dfrac24 \cdot \dfrac13 = \dfrac{2}{4(4-1)}$$