Factorial
In the previous section the same product appeared again and again: $n \cdot (n-1) \cdot (n-2) \cdots 1$. It comes up so often in combinatorics that it has its own name and its own symbol: the factorial. It is worth mastering, because it will go with you through everything that follows.
What is the factorial?
Notice how quickly it grows: going from $6!=720$ to $7!=5\,040$ you only multiply by $7$, but the jump is huge. That is why the numbers in combinatorics shoot up so fast.
The factorial is not arbitrary: it is exactly the number of permutations of $n$ distinct elements that you saw in the previous section, $P_n = n!$.
Calculate the exact value of the expression $6! - 4!$.
Calculate the exact value of the expression $6! - 4!$.
Simplifying fractions of factorials
Calculating $100!$ by hand would be absurd, but often there is no need: when the large factorial is divided by a smaller one, almost everything cancels.
The same trick works in reverse, to write a product as a quotient of factorials:
Simplify the following fraction of factorials without using a calculator: $\frac{8!}{6!}$.
Simplify the following fraction of factorials without using a calculator: $\frac{8!}{6!}$.
Express the numerical product $12 \cdot 11 \cdot 10$ using a division of factorials.
Express the numerical product $12 \cdot 11 \cdot 10$ using a division of factorials.
Factorials with letters
The same technique works when the number is unknown and is written as a letter. In combinatorics exercises it is common to have to simplify expressions such as $\dfrac{n!}{(n-2)!}$.
Simplify the algebraic expression $\frac{(n+1)!}{n!}$, giving the result in terms of $n$.
Simplify the algebraic expression $\frac{(n+1)!}{n!}$, giving the result in terms of $n$.
Simplify the algebraic expression $\frac{n!}{(n-2)!}$ as far as possible, expanding the numerator until terms can be cancelled with the denominator.
Simplify the algebraic expression $\frac{n!}{(n-2)!}$ as far as possible, expanding the numerator until terms can be cancelled with the denominator.
Equations with factorials
If an equation contains factorials of an unknown, the usual approach is to simplify first (as in the previous section) so that you are left with an ordinary equation.
Determine the value of the natural number $x$ given that $x! = 120$.
Determine the value of the natural number $x$ given that $x! = 120$.
Solve the equation $\frac{(x+2)!}{(x+1)!} = 15$, stating the value of $x$.
Solve the equation $\frac{(x+2)!}{(x+1)!} = 15$, stating the value of $x$.