Evaluate permute five five multiplied by permute two two.
Before we actually go on and solve the problem, what I wanna look at first is actually what does this mean. What does permute five five mean? Well, what it actually means is the number of ways five objects can actually be chosen from five groups.
Okay, that’s great. We actually know what it means. But how do we actually calculate it? Well, we actually have this general formula to help us. So we have permute 𝑛 𝑟 is equal to 𝑛 factorial divided by 𝑛 minus 𝑟 factorial. So now we actually have this formula, what we’re actually gonna do is use it to find out each of our values in our original expression.
I’m gonna start with permute five five. So we have 𝑛 is equal to five and 𝑟 is equal to five. So therefore, permute five five is gonna be equal to five factorial divided by five minus five factorial, which will give us five factorial divided by zero factorial. So it’s at this point, we introduce a little relationship which says that zero factorial is just equal to one. So therefore, we can say permute five five is equal to five factorial.
But what does factorial actually mean? Well, the general term is that 𝑛 factorial is equal to the product of all the positive integers less than or equal to 𝑛. So therefore, our five factorial gives us five multiplied by four multiplied by three multiplied by two multiplied by one. So we can say that permute five five is equal to 120.
Great, now, let’s move on to permute two two. Well, for permute two two, we have 𝑛 is equal to two and 𝑟 is equal to two. So we’re gonna substitute these into our formula. So we’re gonna get two factorial divided by two minus two factorial. So we’re gonna get two factorial divided by zero factorial. It’s at this point we remember that zero factorial was equal to one. So we’re gonna have two factorial divided by one which is just equal to two factorial which will give us two multiplied by one. So therefore, permute two two is just equal to two.
Okay, great, so we’ve now found both of our values. What we need to do is multiply them together. So we get permute five five multiplied by permute two two is equal to 120 cause that’s permute five five multiplied by two because that was permute two two, which gives us our final value which is 240.