### Video Transcript

Solve the following equation for
π: five Pπ is equal to 120.

We know that we calculate πPπ by
taking π factorial over π minus π factorial. In this case, we donβt know the
π-value. We know that our set has five
elements, but we donβt know how many weβre trying to choose. To find π, we need to find out how
many decreasing consecutive integers, starting with five, we should multiply
together to equal 120. We know that π factorial is equal
to π times π minus one factorial. Five Pπ is then equal to five
factorial over five minus π factorial. If we expand the factorial in the
numerator, we get five times four times three times two times one. And we know that five Pπ must be
equal to 120, but five factorial equals 120. So we end up with the equation 120
equals 120 over five minus π factorial.

If we multiply both sides of the
equation by five minus π factorial, we get 120 times five minus π factorial equals
120. And then if we divide both sides by
120, on the left, we have five minus π factorial, and on the right, 120 divided by
120 equals one. This means we need an π-value that
will make five minus π factorial equal to one. Based on the properties of
factorials, we know that there are two places where a factorial equals one, zero
factorial and one factorial, which means five minus π must be zero or five minus π
must be one. If five minus π is zero, then π
equals five. And if five minus π is one, then
π equals four.

Remember, at the beginning, we said
that π would be equal to the number of decreasing consecutive integers beginning
with five we multiply together to get 120. And so there is one other strategy
we can use when we know the π-value of a permutation but we donβt know the
π-value. We know the number of permutations
we can have is 120. And we know that we are beginning
with a set of five. If we start by dividing 120 by
five, we get 24. Now we take 24 and we divide by the
consecutive integer below five. So 24 divided by four equals
six.

Again, weβll take that value and
divide it by the integer that is below four. So six divided by three equals
two. Two divided by the integer below
three, which is two, equals one, which shows us that π could equal four, that four
consecutive decreasing integers beginning with five multiply together to equal
120. Two times three times four times
five does equal 120. However, we could follow the
pattern one final time because one divided by one equals one and one times two times
three times four times five also equals 120, which gives us π equals four or π
equals five.