Derivation of the formula for ${ }_{n} P_{r}=\frac{n !}{(n-r) !}$ and ${ }_{n} C_{r}=\frac{n !}{r !(n-r) !}$

Solution 1:

The first formula is simply $n \times n-1 \times \cdots \times n-r+1$ which is correct by the fundamental counting principal. First you pick one of $n$, then you pick one of the remaining $n-1$, and so on.

The second formula is obtained by dividing the first formula by the number of distinct ways of choosing those $r$ terms, since we don't care about the order in which they are selected. The logic used in the first formula tells us this means we should divide by $r!$.

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