Question Video: Evaluating Combinations by Using Properties of Combinations Mathematics

Write ππΆπ + ππΆ(π + 1) in the form ππΆπ.

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Video Transcript

Write π choose π plus π choose π plus one in the form π choose π.

Initially, we might look at these combinations and think we should expand them with the factorial formula to add them together. And while itβs possible to use this method, thereβs a much more straightforward approach. Instead, we can use the recursive property for combinations. This tells us that π minus one πΆπ plus π minus one πΆπ minus one is equal to π choose π.

Now at this point, weβre looking at our combination and thinking, how does this help us since we donβt have any instances of π minus one in our sum of combinations? But we know that weβre looking for something in the form ππΆπ. So first, we convert our recursive property to include πβs and πβs in place of the πβs and πβs. And we have π minus one choose π plus π minus one choose π minus one is equal to π choose π. We can then use this to try and work backwards.

Notice that in the recursive property, weβre adding two combinations with the same set size. Thatβs π minus one. And in one combination, weβre choosing a certain value, thatβs π, and in the other combination, weβre choosing one less than that, π minus one. So now going back to our sum of combinations, we have π plus one and π, which is one less than π plus one. And since π is one less than π plus one, we swap the terms in the recursive property around so that our π corresponds to π minus one. And our π plus one corresponds to π.

Since weβre dealing with addition, itβs fine to change the order and the sum is still equal to π choose π. Doing all this gives us π equal to π minus one. Now if π equals π minus one, adding one to both sides gives π plus one equals π. So with π equal to π plus one and π equal to π plus one, we have π and π in terms of π and π. And this means weβve been able to write the two combinations in our sum as a combination in the form π choose π. So we have π choose π plus π choose π plus one is equal to π plus one choose π plus one.