Permutations, Ordered Counting, and the Multiplication Principle

A bank PIN has 4 digits, so there are $10^4 = 10{,}000$ of them. Drop the repeats and demand 4 distinct digits, and you have only $10 \cdot 9 \cdot 8 \cdot 7 = 5{,}040$. Same alphabet, different rule, half the answers gone. Picking the right ordered-counting rule is the whole game.

Build an outcome in stages. If the first stage can happen $a$ ways and the second $b$ ways regardless of the first, the combined process happens $a \cdot b$ ways. The principle chains across any number of independent stages.

$$N = n_1 \cdot n_2 \cdot n_3 \cdots n_k$$

A meal with 3 appetizers, 5 entrees, and 4 desserts gives $3 \cdot 5 \cdot 4 = 60$ meals. Every permutation rule below is just the multiplication principle applied to a shrinking or fixed menu.

Order all $n$ distinct items in a line. The first slot has $n$ choices, the second has $n-1$ (one item is used up), and so on down to 1.

Common mistakes

• Using $n^r$ when repeats are forbidden. The no-repeat plate count is ${}_{26}P_3 \cdot {}_{10}P_3 = 11{,}232{,}000$, not $26^3 \cdot 10^3 = 17{,}576{,}000$. Trap: 17,576,000 is the with-repetition answer dressed up as the no-repeat one.
• Using ${}_nP_r$ when repeats ARE allowed. A 4-digit PIN with repeats allowed is $10^4 = 10{,}000$, not ${}_{10}P_4 = 5{,}040$. The word "PIN" usually means repeats are fine. Trap: 5,040.
• Using $n!$ for a circular table. Six people around a circle is $(6-1)! = 120$, not $6! = 720$. Rotations are the same seating. Trap: 720.

Bottom line

• Multiplication principle: if step 1 has $a$ outcomes and step 2 has $b$, the pair has $a \cdot b$. Multiply across independent stages.
• Full permutation: order all $n$ distinct items in $n!$ ways, with the convention $0! = 1$.
• Partial permutation (${}_nP_r$): order $r$ of $n$ distinct items in $n!/(n-r)!$ ways. Order matters, no repeats.
• With repetition: fill $r$ ordered slots from $n$ options, repeats allowed, in $n^r$ ways (options to the power slots).

Exam shortcut

Scan the stem for the repetition policy before you pick a formula. "Repeats allowed, digits can repeat, with replacement" means $n^r$; "distinct, no repeats, without replacement" means ${}_nP_r$ (use all items and it collapses to $n!$). For a round table, subtract one from $n$: "circle means $(n-1)!$, because one seat is fixed." When letters repeat in a word, divide $n!$ by the factorial of each repeat count.

The full lesson (about 3,446 words, 23 min read) adds 4 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1a
• 1b
• 1c
• 1d
• 1e
• 1f
• 1g

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