If 𝑛P15 is equal to 23 times 𝑛
minus one P14, find 𝑛.
We have an equation with two
different permutations on either side. On the left, our set is size 𝑛,
and on the right, we have 𝑛 minus one. On the left, we’re choosing 15, and
on the right, we’re choosing 14, which is 15 minus one. We actually have a property of
permutations that fits this pattern. It tells us for 𝑛P𝑟, it’s equal
to 𝑛 times 𝑛 minus one P𝑟 minus one. In our question, we have 𝑛, 𝑛
minus one, then 𝑟, 𝑟 minus one. And this means the value of 𝑛 will
be equal to the coefficient of this other permutation, in our case, 23. And therefore, we can say that 𝑛
equals 23. But you might be wondering, what if
you didn’t remember this property? Is there another way to solve?
If we know that we calculate 𝑛P𝑟
by taking 𝑛 factorial over 𝑛 minus 𝑟 factorial, on the left we have 𝑛 factorial
over 𝑛 minus 15 factorial. And on the right, we have 23 times
𝑛 minus one factorial over 𝑛 minus one minus 14 factorial, where 𝑛 minus one is
in the 𝑛 position and 14 is in the 𝑟 position. We can do a bit of simplifying on
the right so that we have 23 times 𝑛 minus one factorial over 𝑛 minus 15
factorial. Since we have 𝑛 minus 15 factorial
in the denominator on both sides, we can multiply both sides of the equation by 𝑛
minus 15 factorial, which will cancel out these terms. And then, we have 𝑛 factorial is
equal to 23 times 𝑛 minus one factorial.
But we also know the definition of
𝑛 factorial. And that means we’ll substitute for
𝑛 factorial 𝑛 times 𝑛 minus one factorial. We now have an 𝑛 minus one
factorial on both sides of our equation. So, we divide both sides of our
equation by 𝑛 minus one factorial. And that term cancels out on both
sides, leaving us with 𝑛 equals 23.