Video Transcript
If 𝑛 plus one factorial over 𝑛 minus one factorial equals 72, find 𝑛 𝑃 five plus 𝑛 𝑃 six plus 𝑛 𝑃 seven.
We’re trying to find 𝑛 𝑃 five plus 𝑛 𝑃 six plus 𝑛 𝑃 seven. And to do that, we’ll first need to find out what 𝑛 is equal to. And for that, we can use our equation 𝑛 plus one factorial over 𝑛 minus one factorial. If we know that 𝑛 factorial is equal to 𝑛 times 𝑛 minus one factorial, we can use this to rewrite our numerator. We could say that 𝑛 plus one factorial equals 𝑛 plus one times 𝑛 plus one minus one factorial. 𝑛 plus one minus one is 𝑛 plus zero. So we can say that 𝑛 plus one factorial is equal to 𝑛 plus one times 𝑛 factorial. But this doesn’t actually help us simplify what we’ve been given. So we want to try and expand this 𝑛 factorial. And we can do that by saying that 𝑛 factorial is equal to 𝑛 times 𝑛 minus one factorial.
If we bring everything else down, we see that in our numerator and in our denominator, we have the term 𝑛 minus one factorial. We now have 𝑛 plus one times 𝑛 equal 72. And we really have two options to solve for this. We could expand our two terms and see that 𝑛 squared plus 𝑛 equal 72 and then subtract 72 from both sides of our equation, which would give us the quadratic 𝑛 squared plus 𝑛 minus 72 equals zero. Or we could recognize that 𝑛 times 𝑛 plus one means that we’re looking for two consecutive integers that multiply together to produce 72. And then we would just be trying to think of two factors of 72 that are consecutive integers from one another.
We know that eight times nine equals 72, and we can see that eight times eight plus one equals 72. This method does not work very well when we’re dealing with numbers much larger. In that case, it might be helpful to know how to factor. If we wanna factor 𝑛 squared plus 𝑛 minus 72, we need two terms that multiply together to equal negative 72 and add together to equal one. We then get the terms 𝑛 minus eight and 𝑛 plus nine, which means 𝑛 would be equal to eight or 𝑛 could be equal to negative nine. However, remember that our 𝑛-value is going to be a permutation. And since we’re going to be using 𝑛 as a permutation, we would never have a negative number of items for a set. And that means we can ignore the solution 𝑛 equals negative nine.
So we found the first part of our problem: 𝑛 equals eight. And we now know that we need to find eight 𝑃 five plus eight 𝑃 six plus eight 𝑃 seven. We know that to calculate permutations like this, we need 𝑛 factorial over 𝑛 minus 𝑟 factorial. So we have eight factorial over eight minus five factorial plus eight factorial over eight minus six factorial plus eight factorial over eight minus seven factorial, which we can simplify to eight factorial over three factorial plus eight factorial over two factorial plus eight factorial. For our third term, it was eight factorial over eight minus seven factorial, which is one factorial. So eight factorial over one factorial is just equal to eight factorial.
And at this point, it just becomes a matter of simplifying the expression. We could remove a term of eight factorial. Then we have eight factorial times one over three factorial plus one over two factorial plus one, which is equal to eight factorial times one-sixth plus one-half plus one. If you add those together and simplify, you get five-thirds. And if you calculate eight factorial times five-thirds, you get 67,200.
