Question Video: Evaluating Permutations to Find an Unknown Mathematics

If 𝑛P15 = 23((𝑛 − 1) P14), find 𝑛.

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Video Transcript

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.