Video: Finding the Number of Ways to Choose ๐‘› out of ๐‘š Things

The names of 4 students are each written on a piece of paper which are then placed in a hat. If 2 names are randomly selected from the hat, determine the number of all two-student selections that are possible.

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Video Transcript

The names of four students are each written on a piece of paper, which are then placed in a hat. If two names are randomly selected from the hat, determine the number of all two-student selections that are possible.

In this question, weโ€™re looking to choose two names from a selection of four. And so we should be beginning to think about combinations and permutations. Combinations are a way of choosing rearrangements where order doesnโ€™t matter. So, for example, if we were choosing out of the letters ๐ด, ๐ต, ๐ถ, and ๐ท, the arrangement ๐ด and ๐ต would be the same as the arrangement ๐ต and ๐ด. With permutations, order does matter. So ๐ด and ๐ต would be a different arrangement to ๐ต and ๐ด. Letโ€™s think about choosing the two names from a hat. If we choose Ali and Ben, for example, that is exactly the same as choosing Ben and Ali. Order doesnโ€™t matter. And so we can see that we are thinking about the number of combinations.

The number of combinations of size ๐‘Ÿ taken from a collection of ๐‘› items is given by ๐‘›C๐‘Ÿ. And the formula is ๐‘› factorial divided by ๐‘Ÿ factorial times ๐‘› minus ๐‘Ÿ factorial. Weโ€™re choosing two names from a selection of four. So the number of combinations is four C two. And thatโ€™s four factorial over two factorial times four minus two factorial. Now, note that we could use the ๐‘›C๐‘Ÿ button on our calculator. But actually, thereโ€™s a little shortcut that can help us to calculate these by hand. We begin by rewriting four minus two factorial as two factorial. But of course, four factorial is four times three times two times one, which can further be rewritten as four times three times two factorial.

And now we see we can divide through by a common factor of two factorial. In fact, two factorial is simply two, so we can divide once again by two. And we see four choose two is two times three divided by one, which is simply six. There are six ways to choose two names from the four given.

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