Basic Counting

The Art of Counting

Combinatorics answers the question: how many ways can something happen?

Fundamental Rules

Product rule: If task A can be done in $m$ ways and task B in $n$ ways independently, together there are $m \cdot n$ ways.

Sum rule: If task A can be done in $m$ ways OR task B in $n$ ways (mutually exclusive), there are $m + n$ ways total.

Permutations and Combinations

Permutation $P(n, r)$ — ordered arrangements of $r$ items from $n$: $P(n, r) = \frac{n!}{(n-r)!}$

Combination $C(n, r) = \binom{n}{r}$ — unordered selections of $r$ items from $n$: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Arrangements with repetition — $r$ choices from $n$ options with replacement: $n^r$

import math

def permutations(n, r):
    return math.factorial(n) // math.factorial(n - r)

def combinations(n, r):
    return math.factorial(n) // (math.factorial(r) * math.factorial(n - r))

def arrangements_with_repetition(n, r):
    return n ** r

print(permutations(5, 3))   # 60  (5×4×3)
print(combinations(10, 3))  # 120

Your Task

Implement permutations(n, r), combinations(n, r), and arrangements_with_repetition(n, r).