Counting in concrete problems

Formulas are only useful if you can plug them into a problem. Here are the configurations that come up most, and how to recognise them.

Anagrams with repeated letters

How many anagrams of the word ANAGRAMME?

9 letters: A×3, M×2, N, G, R, E

        9!             362,880
   ----------   =   -----------   =  30,240
    3! × 2!             12

The rule: divide n! by the factorial of the count of each repeated letter, since swapping two identical A's does not produce a different word.

Paths on a grid

How many paths from A to B, moving only right or up?

    B
    ^ +---+---+---+
    | |   |   |   |          Each path is a sequence of
    2 +---+---+---+          5 moves: 3 "right" and 2 "up".
    | |   |   |   |
    A +---+---+---+          Choosing the path means choosing which
        3 (right)            2 of the 5 steps go up:  C(5,2) = 10

The whole difficulty of a path problem reduces to this translation: a path is a choice of positions in a sequence.

Simultaneous draws

A simultaneous draw is a draw without order and without replacement — hence a combination.

An urn holds 12 balls: 5 red and 7 black. Three are drawn simultaneously.

number of possible draws       : C(12,3) = 220
draws with exactly 2 red ones  : C(5,2) × C(7,1) = 10 × 7 = 70

                            70
   P(exactly 2 red) = --------- ≈ 0.318
                           220

The pattern to remember: choose separately within each category, then multiply — the multiplication principle again.

Going through the complement

In the same draw, what is the probability of getting at least one red?

directly: exactly 1, 2 or 3 red         ->  three computations
by the complement: no red = 3 black     ->  C(7,3) = 35

                          35
   P(at least one) = 1 - ----- = 1 - 0.159 ≈ 0.841
                         220

Whenever a problem says "at least one", the complement is almost always shorter.

The general method

1. does order matter?          -> arrangement or combination
2. can items repeat?           -> power or factorial
3. are there categories?       -> choose in each, then MULTIPLY
4. does it say "at least"?     -> go through the COMPLEMENT

A plausibility check

A count can often be checked by order of magnitude: a result must stay below the total number of cases, and a probability must fall between 0 and 1. A sum of probabilities exceeding 1 almost always signals double counting — the same case counted in two categories.

Summary

• Anagrams: n! divided by the factorial of the repeated counts.
• Grid paths: choose the "up" steps among the total → a combination.
• Simultaneous draw: no order, no replacement → C(n,k).
• Several categories: choose in each, then multiply.
• "At least one": go through the complement.
• A probability above 1 betrays double counting.