Video: Using Properties of Combinations to Find the Value of a Given Ratio

Determine the value of ²¹𝐶₇/²¹𝐶₆ without using a calculator.

03:49

Video Transcript

Determine the value of 21 choose seven over 21 choose six without using a calculator.

To help us calculate the value of 21 choose seven and 21 choose six, what we actually have is a general formula. And that formula is that if we have 𝑛 choose 𝑟, then this is equal to 𝑛 factorial over 𝑟 factorial multiplied by 𝑛 minus 𝑟 factorial. But we can see here that we’ve actually got factorials.

So just a quick recap so to understand what a factorial is. For instance, if I have 𝑛 factorial, then what this actually means is the product of all positive integers that are less than or equal to 𝑛. For example, if we have four factorial, this would be equal to four multiplied by three multiplied by two multiplied by one.

Okay, great! So now we know what the general formula is to find out what 𝑛 choose 𝑟 is. And we also have reminded ourselves what a factorial is. We can get on and determine the value of 21 choose seven over 21 choose six.

And we’ll start with working out the value of 21 choose seven. And this is gonna be equal to 21 factorial over seven factorial multiplied by 21 minus seven factorial. And that’s because our 𝑛 is 21 and our 𝑟 is seven. And this is all gonna be equal 21 factorial divided by seven factorial multiplied by 14 factorial. And we’ll leave it in this form at the moment because this is a question without a calculator. So therefore, if we leave it at this stage, we can do that without having to use any further calculations.

Okay, now let’s move on to 21 choose six. So 21 choose six is gonna be equal to 21 factorial over six factorial multiplied by 21 minus six factorial. So this is gonna be equal to 21 factorial over six factorial multiplied by 15 factorial.

Okay, fantastic! We now have the two values that we want. So now what we need to do is actually go back to the question and see if we can find out what 21 choose seven divided by 21 choose six is gonna be equal to.

So we’re gonna get 21 factorial over seven factorial multiplied by 14 factorial divided by 21 factorial over six factorial multiplied by 15 factorial. Well, we know that, actually, from the division rule of fractions that if we’re gonna divide by a fraction, it’s the same as multiplying by the reciprocal of the same fraction.

So now actually we have it in this form. We can start to simplify. So first of all, we can actually divide the numerator and denominator by 21 factorial. So therefore, we can cancel these out, which means we’re left with six factorial multiplied by 15 factorial over seven factorial multiplied by 14 factorial.

Well, first of all, if we take a look at 15 factorial over 14 factorial, we see that 15 factorial is 15 times 14 times 13 etc. And 14 factorial is 14 times 13 times 12 etc. So therefore we can actually divide the numerator and the denominator by our 14 factorial. And we’re just left with 15 on our numerator.

So now let’s have a look at six factorial over seven factorial. We’ve got six times five times four etc. over seven times six times five times four etc. So therefore, like the previous example, we can actually divide the numerator and the denominator by six factorial. So then we’re just left with seven as a denominator.

So actually, the value of six factorial over seven factorial would be one-seventh. So it’s gonna be equal to one over seven multiplied by 15. So therefore, we can say that the value of 21 choose seven divided by 21 choose six is 15 over seven.

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