Probability and mathematical statistics - Combinatorics

Classical probability theory definition of $P(A)$ – where $A$ is success event – is computed by following formula:

$$P(A) = \frac{N(A)}{N(S)}$$

where $N(A)$ denotes the number of elements of $A$ and $N(S)$ denotes the number of elements in the sample space $S$.

Permutation

$$P_n = n!$$

Variation (combinatorics)

Variation with repetition:

$$\bar{A}_u^o = k^n$$

Variation without repetition:

$$A_u^o = \frac{n!}{(n-k)!}$$

Combination

$$C_n^k = \frac{n!}{k!(n-k)!} = \dbinom{n}{k}$$

We could also compute

$$C_n^k \cdot P_k = \frac{n!}{(n-k)!} = A_n^k$$