### Video Transcript

In this video, we will learn how to
find the factorial of any positive integer 𝑛 which is the product of all integers
less than or equal to 𝑛 and greater than or equal to one. We will then consider how we can
use factorials to solve problems. We will begin by looking at the
definition of a factorial.

The factorial of a positive integer
𝑛 is the product of all the positive integers less than or equal to 𝑛. We use either of the notations
shown, which are both read as 𝑛 factorial. And this is equal to 𝑛 multiplied
by 𝑛 minus one multiplied by 𝑛 minus two and so on multiplied by two multiplied by
one.

We also define the factorial of
zero to be equal to one. That is, zero factorial is equal to
one. For any integer 𝑛 such that 𝑛 is
greater than or equal to one, then 𝑛 factorial is also equal to 𝑛 multiplied by 𝑛
minus one factorial. We can see this from the general
definition, as 𝑛 minus one multiplied by 𝑛 minus two and so on multiplied by two
multiplied by one is equal to 𝑛 minus one factorial. It is this property that is
particularly important when solving complex problems involving factorials. However, in our first example, we
will practice computing a factorial.

Evaluate four factorial.

We begin by recalling that the
factorial of any positive integer 𝑛 is the product of all the positive integers
less than or equal to 𝑛. This means that 𝑛 factorial is
equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two and so on
multiplied by two multiplied by one. Four factorial is therefore equal
to four multiplied by three multiplied by two multiplied by one.

As multiplication is commutative,
we can multiply our integers in any order. For example, four multiplied by
three is equal to 12. Two multiplied by one is equal to
two. And then multiplying 12 by two
gives us an answer of 24. Alternatively, we could multiply
four by three, giving us 12, then multiply this by two, giving us 24, and finally
multiplying 24 by one gives us a final answer of 24. This confirms that four factorial
is equal to 24.

Our next example is a more
complicated problem involving factorials.

Simplify the expression six
factorial over four factorial minus 27 factorial over 28 factorial. Give you answer as a fraction.

We recall that since 𝑛 factorial
is the product of all the positive integers less than or equal to 𝑛, then 𝑛
factorial is equal to 𝑛 multiplied by 𝑛 minus one multiplied by 𝑛 minus two and
so on all the way down to one. This leads us to a key property of
factorials. 𝑛 factorial is equal to 𝑛
multiplied by 𝑛 minus one factorial. This means that instead of
calculating each of the factorials individually, we can simplify and look for common
factors.

We can write six factorial as six
multiplied by five multiplied by four factorial. This means that the first term in
our expression can be written as six multiplied by five multiplied by four factorial
divided by four factorial. We can then cancel a factor of four
factorial from the numerator and denominator, leaving us with six multiplied by
five.

In the same way, we can write the
denominator of our second term as 28 multiplied by 27 factorial. This time, we can cancel the factor
of 27 factorial on the numerator and denominator. And this term simplifies to one
over 28. Six multiplied by five is equal to
30. So we need to subtract one
twenty-eighth from 30. We could write this as a mixed
number as 29 and twenty-seven twenty-eighth. However, as we want to give our
answer just as a fraction, we will convert the number 30 or 30 over one to a
fraction over 28.

In order to do this, we will
multiply 30 by 28. As 28 multiplied by three is 84, 28
multiplied by 30 is 840. This means that 30 or 30 over one
is equal to 840 over 28. As our denominators are the same,
we simply subtract the numerators. And we can therefore conclude that
six factorial over four factorial minus 27 factorial over 28 factorial written as a
fraction is 839 over 28.

In our final example, we will use
our knowledge of factorials to solve an algebraic equation.

Find the solution set of one over
𝑛 plus seven factorial plus one over 𝑛 plus eight factorial is equal to 256 over
𝑛 plus nine factorial.

In order to answer this question,
we begin by recalling that for any integer 𝑛 greater than or equal to one, then 𝑛
factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. The fact that 𝑛 must be greater
than or equal to one will be important when finding the solution set.

When dealing with a problem where
the sum of two fractions is equal to another fraction, it is often useful to try and
eliminate the denominators first. In this question, we will multiply
all three terms by 𝑛 plus nine factorial. This gives us 𝑛 plus nine
factorial over 𝑛 plus seven factorial plus 𝑛 plus nine factorial over 𝑛 plus
eight factorial is equal to 256 multiplied by 𝑛 plus nine factorial over 𝑛 plus
nine factorial. On the right-hand side of our
equation, we can cancel the common factor of 𝑛 plus nine factorial, leaving us with
256.

Using the fact that 𝑛 factorial is
equal to 𝑛 multiplied by 𝑛 minus one factorial and it is also equal to 𝑛
multiplied by 𝑛 minus one multiplied by 𝑛 minus two factorial, we can rewrite the
left-hand side of our equation as shown. In the first term, we can cancel a
factor of 𝑛 plus seven factorial. And in the second term, we cancel a
factor of 𝑛 plus eight factorial. We now have an equation that no
longer contains any fractions. 𝑛 plus nine multiplied by 𝑛 plus
eight plus 𝑛 plus nine is equal to 256.

We can simplify the left-hand side
either by taking out a factor of 𝑛 plus nine or by using the FOIL method to
distribute our parentheses. 𝑛 plus nine multiplied by 𝑛 plus
eight is equal to 𝑛 squared plus eight 𝑛 plus nine 𝑛 plus 72. And when we add 𝑛 plus nine to
this, we get an answer of 256. We can then group or collect the
like terms on the left-hand side. This gives us 𝑛 squared plus
18𝑛. And when we subtract 256 from both
sides, we have negative 175. We now have a quadratic equation 𝑛
squared plus 18𝑛 minus 175 is equal to zero.

Our next step is to factor the
expression on the left-hand side into two sets of parentheses. Since the coefficient of 𝑛 squared
is equal to one, we know that the first term in each of them will be 𝑛. And the second terms will have a
sum of positive 18 and a product of negative 175. One factor pair of 175 is 25 and
seven. This means that multiplying 25 by
negative seven gives us negative 175. And since 25 plus negative seven is
18, our two sets of parentheses are 𝑛 plus 25 and 𝑛 minus seven.

As the product of 𝑛 plus 25 and 𝑛
minus seven equals zero, then either 𝑛 plus 25 equals zero or 𝑛 minus seven equals
zero. This gives us two possible
solutions 𝑛 equals negative 25 and 𝑛 equals seven. As already mentioned, we know that
𝑛 must be greater than or equal to one, as factorials are only defined for
nonnegative integers. The value of 𝑛 that satisfies the
equation is therefore equal to seven. And the solution set of the
equation one over 𝑛 plus seven factorial plus one over 𝑛 plus eight factorial is
equal to 256 over 𝑛 plus nine factorial contains the number seven.

We will now summarize the key
points from this video.

The factorial of a positive integer
𝑛 is defined as the product of all positive integers less than or equal to 𝑛 such
that 𝑛 factorial is as shown. The key property of the factorial
is that 𝑛 factorial is equal to 𝑛 multiplied by 𝑛 minus one factorial. And we can use this to simplify
expressions involving factorials. Whilst it was not covered by a
specific example in this video, when trying to find an unknown integer given its
factorial, we divide by consecutive positive integers.

This means that in order to find
the value of 𝑛 such that 𝑛 factorial is equal to 120, we divide by the integers
one, two, three, and so on. 120 divided by one is 120. Dividing this by two gives us
60. Dividing 60 by three gives us
20. 20 divided by four is equal to
five. And finally, five divided by five
is equal to one. We can therefore conclude that 120
is equal to five multiplied by four multiplied by three multiplied by two multiplied
by one, which can be written as five factorial. The value of 𝑛 such that 𝑛
factorial is 120 is five.