### 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, weโre 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 doesnโt matter. So, for example, if we were
choosing out of the letters ๐ด, ๐ต, ๐ถ, and ๐ท, the arrangement ๐ด and ๐ต would be
the same as the arrangement ๐ต and ๐ด. With permutations, order does
matter. So ๐ด and ๐ต would be a different
arrangement to ๐ต and ๐ด. Letโs 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 doesnโt matter. And so we can see that we are
thinking about the number of combinations.

The number of combinations of size
๐ taken from a collection of ๐ items is given by ๐C๐. And the formula is ๐ factorial
divided by ๐ factorial times ๐ minus ๐ factorial. Weโre choosing two names from a
selection of four. So the number of combinations is
four C two. And thatโs four factorial over two
factorial times four minus two factorial. Now, note that we could use the
๐C๐ button on our calculator. But actually, thereโs 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.