Question Video: Finding an Unknown by Evaluating Permutations Mathematics • Second Year of Secondary School

Video Transcript

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

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

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

So, we have a statement that says 𝑥 times 𝑥 minus one equals three factorial. 𝑥 times 𝑥 minus one equals 𝑥 squared minus 𝑥. 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 𝑥 squared minus 𝑥 minus six. And we’re 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 𝑥, 𝑥 plus two and 𝑥 minus three, which means 𝑥 should either be negative two or positive three.

However, we need to think carefully about this. When we’re dealing with permutations, the 𝑛 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 𝑥 cannot equal negative two, but 𝑥 does equal negative three. Now, we should be careful here. That’s not our answer. We are looking for 𝑛. And 𝑥 equals 𝑛 minus 10. So, we can say that three equals 𝑛 minus 10. Adding 10 to both sides, we see that 13 equals 𝑛 or, more commonly, 𝑛 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 𝑛 equals 13.

Before we move on, I wanna go back to this step. When we had 𝑥 times 𝑥 minus one equals three factorial and we know that three factorial equals six, we could interpret 𝑥 times 𝑥 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, 𝑥 equals six. This strategy uses logic and knowledge about what’s happening in permutations to find the value of 𝑥, whereas the first method we used was a more algebraic method. And either method will help us get to the final answer that 𝑛 equals 13.