Evaluate six permute three plus seven permute five.
First of all, what does six permute three actually mean? So what six permute three or permute six three means is the number of ways six objects can be chosen from three groups. Okay, so now we know what it is. How do we actually calculate it? But we actually have a general formula to help us calculate the value. And we can say that 𝑛 permute 𝑟 is equal to 𝑛 factorial over 𝑛 minus 𝑟 factorial.
So this point there, it might be worth recapping what factorial means. Well, for example, if we have four factorial, this is actually equal to four multiplied by three multiplied by two multiplied by one. So what it’s actually is, is all the positive integers that are the same or less than the value that we have the factorial of multiplied together. Okay, so now we know what factorial is. And now we’ve got our general form. Let’s use it to evaluate our expression.
So first of all, I’m gonna work out the values of six permute three and seven permute five. So six permute three is equal to six factorial over six minus three factorial which is equal to six factorial over three factorial. And then, I’m just gonna show this next line. I mean you could usually put six factorial over three factorial and calculate it to get the value. But I just wanna show where it comes from. So six factorial is six multiplied by five multiplied by four multiplied by three multiplied by two multiplied by one. And three factorial, three by two by one.
Well, we could see that as we have three multiplied by two multiplied by one on the numerator and denominator, they would actually cancel. So we’re left with six multiplied by five multiplied by four. So therefore, it gives us an answer of 120. Okay, so now we found this. Let’s move on to seven permute five.
Well, seven permute five is gonna be equal to seven factorial over seven minus five factorial which is equal to seven factorial over two factorial. So this is gonna give us the answer of 2520. So fantastic, we’ve now got six permute three and seven permute five. Let’s move on to the final stage.
Well, the final stage is to actually evaluate six permute three plus seven permute five. So we’re gonna have 120 plus 2520 which gives us a final answer for six permute three plus seven permute five of 2640.