### Video Transcript

Find the value of π given that π
minus 10P two is equal to three factorial.

First of all, letβs think about
what we know about permutations. For the permutation πPπ, it will
be equal to π factorial over π minus π factorial. In the π position in our
permutation, we have this expression π minus 10. When we have an expression for our
π position, it can be a good idea to substitute a variable π, which will make the
calculations simpler. So, we will say let π₯ equal π
minus 10. We now have something that says π₯P
two equals three factorial. And then, we can write our
permutation as π₯ factorial over π₯ minus two factorial equals three factorial.

We wanna try and simplify our
permutation. And we can do that with a property
of factorials, π factorial is equal to π times π minus one factorial, which means
that π₯ factorial can be rewritten as π₯ times π₯ minus one factorial and π₯ minus
one factorial can be rewritten as π₯ minus one times π₯ minus two factorial. By expanding our numerator in this
way, we end up with the term of π₯ minus two factorial in the numerator and the
denominator, and they cancel out.

So, we have a statement that says
π₯ times π₯ minus one equals three factorial. π₯ times π₯ minus one equals π₯
squared minus π₯. Three factorial is three times two
times one, which is six. It now looks like we have some kind
of quadratic. If we subtract six from both sides
of our equation, we get π₯ squared minus π₯ minus six. And weβre gonna solve that by
factoring. We need the factors that multiply
together to equal negative six and add together to equal negative one. That will be positive two and
negative three. So, we have two terms for π₯, π₯
plus two and π₯ minus three, which means π₯ should either be negative two or
positive three.

However, we need to think carefully
about this. When weβre dealing with
permutations, the π position represents the number of items in a set. And we would never say that we have
negative items in a set. And that means π₯ cannot equal
negative two, but π₯ does equal negative three. Now, we should be careful here. Thatβs not our answer. We are looking for π. And π₯ equals π minus 10. So, we can say that three equals π
minus 10. Adding 10 to both sides, we see
that 13 equals π or, more commonly, π equals 13. What we have here is 13 minus 10P
two equals three factorial, which would be three P two equals three factorial. And that is a true statement, which
confirms π equals 13.

Before we move on, I wanna go back
to this step. When we had π₯ times π₯ minus one
equals three factorial and we know that three factorial equals six, we could
interpret π₯ times π₯ minus one to say these are two consecutive integers that must
multiply together to equal six. And since three and two are the
factors of six and three and two are consecutive integers, we could have used that
knowledge to say that three times three minus one equals six. Therefore, π₯ equals six. This strategy uses logic and
knowledge about whatβs happening in permutations to find the value of π₯, whereas
the first method we used was a more algebraic method. And either method will help us get
to the final answer that π equals 13.