How many ways can we arrange the letters in the word MATH ?
Imagine we have four blank spaces that represent where each letter in the word MATH can go.
_ _ _ _
From here we can ask ourselves, how many choices of the letters in the word MATH can go in the first position?
Since there are 4 letters, there are 4 options; the arrangement can begin with M, A, T, or H.
Now let's imagine we chose one of those letters and locked it into the first position.
X _ _ _
or we can place a 4 here to represent the four options:
4 _ _ _
Now if we look to the next blank, how many options do we now have for the second position?
Since we already used one letter for the first blank, we are now left with 3 letters we can choose from.
X X _ _
or 4 3 _ _
If we continue and lock in one of the three letters into the second position, that leaves us with 2 options for the third position and subsequently one letter for the fourth position.
4 3 2 1
So how can we use this to figure out the number of unique arrangements?
Well, you can imagine this as an application of the product rule. We are choosing one letter for the first position AND one letter for the second position AND one letter for the third AND one letter for the fourth. Since we figured out how many options we have for each of those choices, we can use the product rule to find the number of arrangements.
\(4 \times 3 \times 2 \times 1 = 24\)
Then there are 24 arrangements. We call these arrangements permutations which we will talk more about in the next lesson.
This product where each term is decreasing by 1 until we reach 1 has a special name; we call this a factorial.
We represent a factorial with an exclamation mark like this:
\(4 \times 3 \times 2 \times 1 = 4!\)
Some things to note:
\(1! = 1\)
\( (n+1) \times n! = (n+1)! \)
For example:
\(5 \times 4! = 5!\)
or
\(90 \times 8! = 10 \times 9 \times 8! = 10!\)