Factorials

Summary: The factorial $n!$ counts the number of distinct orderings (permutations) of $n$ different objects; dividing by sub-factorials adjusts the count when some objects are identical, yielding multinomial and binomial coefficients.

Sources: chapter-1.2.11

Last updated: 2026-04-29

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Definition

For a positive integer $n$, the factorial is

$n!=n\times (n-1)\times (n-2)\times \cdots \times 2\times 1$

Euler does not use the symbol $n!$ (that notation came later), but he builds the concept explicitly in ch1.2.11-transpositions-and-binomial-proof to justify the binomial-theorem. (source: chapter-1.2.11)

Counting Permutations

The factorial gives the number of ways to arrange $n$ distinct objects in a row:

• 1 object: $1!=1$ arrangement
• 2 objects: $2!=2$ arrangements
• 3 objects: $3!=6$ arrangements (each of 3 can be first; the remaining 2 admit 2 arrangements)
• 4 objects: $4!=24$; and so on.

The recurrence is $n!=n\times (n-1)!$ because each of the $n$ objects can be first, leaving $(n-1)!$ arrangements for the rest. (source: chapter-1.2.11)

Repeated Objects

If a group of $n$ objects contains $p$ identical copies of one type and $q$ identical copies of another (with $p+q\le n$), the number of distinct arrangements is

$\frac{n!}{p!q!\cdots }$

Example: the letters $aaabbc$ have $6!/(3!\cdot 2!\cdot 1!)=60$ distinct orderings. (source: chapter-1.2.11)

Binomial Coefficients

A term $a^{n-k}{b}^k$ in the expansion of $(a+b{)}^n$ consists of $n$ letters with $n-k$ identical $a$‘s and $k$ identical $b$‘s. Its coefficient is

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$

This is the entry at position $k$ (zero-indexed) in row $n$ of pascal-triangle. (source: chapter-1.2.11)

Multinomial Coefficients

For a term $a^p{b}^q{c}^r\cdots$ in the expansion of $(a+b+c+\cdots {)}^n$ with $p+q+r+\cdots =n$, the coefficient is

$\frac{n!}{p!q!r!\cdots }$

Euler applies this to $(a+b+c{)}^3$, obtaining coefficient 6 for the mixed term $abc$ (since $3!/(1!\cdot 1!\cdot 1!)=6$). (source: chapter-1.2.11)

Role in the Book

Factorials arise here as a combinatorial tool to prove why binomial coefficients have the values they do. They do not appear elsewhere in Elements of Algebra (which is an introductory text), but they are foundational to combinatorics, probability, and the theory of infinite-series via the binomial-theorem.

Related pages

• binomial-theorem
• pascal-triangle
• ch1.2.11-transpositions-and-binomial-proof
• algebraic-identities