The portal has been deactivated. Please contact your portal admin.

Question Video: Evaluating Permutations to Find an Unknown Mathematics

If 𝑛P15 = 23((𝑛 βˆ’ 1) P14), find 𝑛.

02:33

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 𝑛Pπ‘Ÿ, it’s equal to 𝑛 times 𝑛 minus one Pπ‘Ÿ minus one. In our question, we have 𝑛, 𝑛 minus one, then π‘Ÿ, π‘Ÿ 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 𝑛Pπ‘Ÿ by taking 𝑛 factorial over 𝑛 minus π‘Ÿ 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 π‘Ÿ 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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.