Determine the value of 23 choose eight divided by 23 choose six without using a calculator.
First of all, we’re just going to look at the notation that we’re using. So what does this actually mean? And what it actually means is the number of different combinations of six objects that can be made from 23 objects. So that’s kind of what we’re talking about looking at combinations here. Okay, so now let’s get on and determine the value.
So we actually have a general formula to help us. So we can say that 𝑛 choose 𝑟 is equal to 𝑛 factorial divided by 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial. And just reminding ourselves that actually 𝑛 factorial anything factorial is the product of all positive integers less than or equal to in this case 𝑛 or that number. So for instance, three factorial would be three multiplied by two multiplied by one.
Okay, great, so we have our notation. We have our formula. Let’s get on and determine the value of 23 choose eight divided by 23 choose six. And we’ll start by working out the value of the numerator and then the value of the denominator. So this is going to be equal to 23 factorial because that’s like our 𝑛 value divided by eight factorial. And this is gonna be multiplied by 23 factorial minus eight factorial.
Okay, now let’s start to simplify. So we get 23 factorial over eight factorial multiplied by 15 factorial. And we’re actually gonna leave this at that stage and then we’ll come back to this actually when we get to the final stage of determining the value of 23 choose eight divided by 23 choose six.
So now, we have 23 choose six is equal to 23 factorial divided by six factorial multiplied by 23 factorial minus six factorial. So this is gonna be equal to 23 factorial divided by six factorial multiplied by 17 factorial. So now, we have the values for each term in our expression. Let’s get on and determine the value.
So first of all, we’re going to have 23 factorial divided by eight factorial multiplied by 15 factorial because that’s our 23 choose eight. And then, this is going to be divided by 23 factorial divided by six factorial multiplied by 17 factorial because this was our 23 choose six.
So we actually have two fractions here. So when we divide fractions the way that we actually do this is we actually find a reciprocal. So we actually swap the numerator and denominator and we multiply. So therefore, this is equal to 23 factorial and then multiplied by six factorial multiplied by 17 factorial because that was the denominator of our original 23 choose six. And then, this is all divided by eight factorial multiplied by 15 factorial multiplied by 23 factorial.
As we can see here, we can actually start to cancel values out because we’ve got 23 factorial on the numerator and 23 factorial on the denominator. So if we divide both by that, it cancels out the 23 factorial. And then next, we can look at the 17 factorial divided by 15 factorial because actually what we can see is on the top because we have 17 factorial 17 multiplied by 16 multiplied by 15 multiplied by 14 et cetera. But on the bottoms on our denominator, we have 15 factorial which is 15 multiplied by 14 multiplied by 13 et cetera. So therefore, if we divide the top and bottom by the 15 factorial, we’re just left with 17 multiplied by 16.
Okay, so now, let’s move on to our final part, which is gonna be six factorial divided by eight factorial. Well, if we think about this, this is gonna to be equal to six multiplied by five multiplied by four multiplied by three et cetera divided by eight multiplied by seven multiplied by six multiplied by five multiplied by four et cetera. So therefore, if we divide top and bottom by six factorial, we’d just left with eight multiplied by seven on our denominator.
Okay, so as we now have 17 multiplied by 16 over eight multiplied by seven, we could work this out. However, as it says do this without using a calculator, we can actually do one more step, which makes life a lot easier. We can actually divide the numerator and denominator both through by eight. So we get 17 multiplied by two divided by and then it would just be one multiplied by seven. But we’re happy with just saying seven. So it’s 17 multiplied by two over seven, which gives us our final answer of 34 over seven. So therefore, we can say the value of 23 choose eight divided by 23 choose six is 34 over seven.