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Evaluate ⁸𝐶₅ : ⁴𝐶₂.
Evaluate the ratio eight choose five, four choose two.
To help us actually evaluate this ratio, what we have is a general form. So if we have 𝑛 choose 𝑟, this is equal to 𝑛 factorial over 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial. And to remind us what actually factorial means, well 𝑛 factorial means the product of all positive integers less than or equal to 𝑛.
For example, we have five factorial. And this is equal to five multiplied by four multiplied by three multiplied by two multiplied by one. Okay, great! So now we have our formula and we also know what factorial means, let’s get on and evaluate our ratio. So I’m gonna start the left-hand side, which is eight choose five.
And this is going to be equal to eight factorial, cause that’s like our 𝑛, over five factorial, our 𝑟, multiplied by eight minus five factorial, which is gonna give us eight factorial over five factorial multiplied by three factorial. It’s at this stage we can actually look at simplifying, because actually we can think, “Right, what’s eight factorial divided by five factorial?”
Well if we think about it, eight factorial is eight multiplied by seven multiplied by six multiplied by five multiplied by four, et cetera. And five factorial is five multiplied by four multiplied by three, et cetera. So if we’re actually gonna to divide the numerator by the denominator, so eight factorial by five factorial, so then what we’re left with as our numerator is eight multiplied by seven multiplied by six multiplied by well one.
And that’s because if we have five multiplied by four multiplied by three multiplied by two multiplied by one divided by five factorial, well in fact they’re the same thing so that would actually be one. And that means that we’ll cancel out the values on our denominator.
Okay, great! So now we’ve got this. We can actually substitute it back into our equation. So this is going to be equal to eight multiplied by seven multiplied by six for the reasons we’ve said, then all divided by — and I’ve expanded three factorial — so we’ve got three multiplied by two multiplied by one. So now we can actually look at simplifying this.
So now we can actually cancel some values out because we’ve got six divided by three, which is just two, so we’ve got that on our numerator. And then eight divided by two just give us four. So we’re left with four multiplied by seven multiplied by two over one, which is just 56. Okay, great! We’ve now found eight choose five. Let’s move on to four choose two, the right-hand side of our ratio.
So for four choose two, we have four factorial over two factorial multiplied by four minus two factorial, which will give us four factorial over two factorial multiplied by two factorial, which would just give us four times three over two factorial. And the reason we got that is like we did for the left-hand side of the ratio.
It’s because actually we divided some factorials. So I’ll show you how we’ve done that. We have four multiplied by three multiplied by two multiplied by one, cause that’s four factorial, over two multiplied by one times two multiplied by one because that’s we got two two factorials on the denominator. So therefore, we can cancel out because we have two multiplied by one on our numerator.
And therefore if we divide that by two factorial, which is two multiplied by one, we can actually cancel that out. So we have one on each the numerator and denominator. So we’re just left with four multiplied by three over two multiplied by one. So we just get 12 over two multiplied by one, so 12 over two. So therefore four choose two is equal to six.
Okay, great! So now let’s go on to the final stage of the evaluation. So if we have the ratio eight choose five, four choose two, then we’re gonna have the ratio 56 to six. And we can actually simplify this a bit further to get the ratio 28 to three. So therefore, we’ve arrived at our final answer and can say that eight choose five, four choose two is equal to 28, three.
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