Permutations, Ordered Counting, and the Multiplication Principle | Free Exam P Lesson

A bank PIN has 4 digits, so there are $1 0^{4} = 10, 000$ 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.

$N = n_{1} \cdot n_{2} \cdot n_{3} \dots n_{k}$

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.

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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 $2 6^{3} \cdot 1 0^{3} = 17, 576, 000$ . Trap: 17,576,000 is the with-repetition answer dressed up as the no-repeat one.
Using $_{n} P_{r}$ when repeats ARE allowed. A 4-digit PIN with repeats allowed is $1 0^{4} = 10, 000$ , 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 ( $_{n} P_{r}$ ): 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 $_{n} P_{r}$ (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 3,031 words, 20 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

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