Question Video: Evaluating Combinations Using Their Properties Mathematics • 12th Grade

Evaluate ¹⁵⁷𝐶₁₅₇ + ⁵¹𝐶₁.

03:16

Video Transcript

Evaluate 157 choose 157 plus 51 choose one.

Now, to help us solve this problem, we actually have a general form. And that is 𝑛 choose 𝑟 is equal to 𝑛 factorial over 𝑟 factorial 𝑛 minus 𝑟 factorial. And we can actually use this to help us find the value of our terms. Well, when we look to evaluate 157 choose 157 plus 51 choose one, what we need to do is actually deal with each of these terms separately. So we’ll just start with 157 choose 157. And this is actually in the form 𝑛 choose 𝑛 because actually our 𝑛 value is actually the same as our 𝑟 value.

And there’s actually a special relationship that tells us that if we have 𝑛 choose 𝑛, then this is equal to one. So therefore, we can actually say that our 157 choose 157 is also gonna be equal to one. But what I’m gonna do is actually prove this by showing how it’ll work if we use the general form. So if we have 157 choose 157, then this is gonna be equal to 157 factorial, because that’s our 𝑛, over 𝑟 factorial, which is 157 factorial, then multiplied by 𝑛 minus 𝑟, so 157 minus 157, factorial. And this is gonna be equal to 157 factorial over 157 factorial multiplied by zero factorial. Well one of the rules we have is that zero factorial is equal to one. So therefore, we’re gonna have this all equal to 157 factorial over 157 factorial which is equal to one. And so that proves what we said at the beginning which was 𝑛 choose 𝑛 — or in this case, 157 choose 157 — is equal to one.

Okay, brilliant. Now, let’s move on and evaluate our next term. So if we look at our second term which is 51 choose one, then yet again, this actually has a special form because if we have 𝑛 choose one, then this is always equal to 𝑛. But, we’re gonna use the general formula just to show this again. So we get 51 choose one is equal to 51 factorial over one factorial multiplied by 51 minus one factorial which is gonna be equal to 51 factorial over one factorial multiplied by 50 factorial. Well, we also know that one factorial is equal to one. So therefore, we’re gonna have 51 factorial over 50 factorial.

Well, if we think about what this actually means, it’ll be 51 times 50 times 49 times 48 all divided by 50 times 49 times 48, et cetera. So therefore, if we actually divide through by our 50 factorial, we’re just left with 51. So therefore, we can say that 51 choose one gives us an answer of 51. And great, this actually fits what we thought had happened at the beginning because we said that 𝑛 choose one is equal to 𝑛. Well, 𝑛 was 51 in our case. So 51 choose one is equal to 51. Okay, so now what we need to do is actually evaluate 157 choose 157 plus 51 choose one. So it’s gonna be equal to one plus 51. So therefore, we can say that if we evaluate 157 choose 157 plus 51 choose one we get 52.