Video Transcript
If three 𝑛 minus five factorial equals 24, find 𝑛 choose 𝑛 minus one.
Now, this might seem like quite a complicated equation to solve, but we do have a bit of an advantage. We know that we can only find the factorial of integer values. So, first of all, three 𝑛 minus five must be an integer. And so, let’s let 𝑥 be equal to three 𝑛 minus five. We want to find the value of 𝑥 such that 𝑥 factorial is equal to 24.
We might recognize this or since it’s quite a small number, we can use trial and error. Let’s try three factorial. Three factorial is three times two times one, which is equal to six. So, 𝑥 is not equal to three. Let’s try four factorial then. Four factorial is four times three times two times one, which is equal to 24. And so, this tells us that 𝑥 must be equal to four. But remember, we let 𝑥 be equal to three 𝑛 minus five. So, we can say that this means that three 𝑛 minus five must be equal to four.
Let’s solve for 𝑛 by adding five to both sides of our equation to give us three 𝑛 equals nine. Next, we’ll divide through by three. And we find 𝑛 is equal to three. At this stage, we could go back to our original expression and check that three 𝑛 minus five factorial is equal to 24 when 𝑛 is equal to three. Well, three 𝑛 minus five factorial is three times three minus five factorial, which is four factorial. And we already saw that was equal to 24. So, we know we’ve chosen the correct value of 𝑛.
Now, the question wants us to find 𝑛 choose 𝑛 minus one. We’ve just evaluated 𝑛 to be equal to three. So, 𝑛 choose 𝑛 minus one must be three choose two. And so, next, we recall what we actually mean by 𝑛 choose 𝑟 or 𝑛𝐶𝑟. It’s 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. So, letting 𝑛 be equal to three and 𝑟 be equal to two, we find that 𝑛 choose 𝑛 minus one is three factorial over two factorial times three minus two factorial or three factorial over two factorial times one factorial.
But, of course, three factorial, which is three times two times one, can be written as three times two factorial. And then, we can see that we can divide both the numerator and the denominator by two factorial. Finally, one factorial is simply equal to one. And so, we get three divided by one, which is equal to three. And so, if three 𝑛 minus five factorial is equal to 24, then 𝑛 choose 𝑛 minus one is equal to three.
