### Video Transcript

Which of the following is equal to
ππΆ one times six ππΆ six over seven ππΆ seven? (A) Six ππ six over seven π
minus seven π seven. (B) Six ππ six over seven π
minus one π six. (C) Six ππ six over seven π
minus seven π six. (D) Six ππ six over seven π
minus one π seven.

Almost immediately, we notice that
our initial expression is in terms of combinations, three different combinations,
while all of our answer choices are given as an expression in terms of
permutations. So we need to rewrite these
combinations in terms of permutations. But first, we notice this ππΆ one,
and the properties of combinations tell us that ππΆ one just equals π. To choose one item from a set of
π, there are π different ways to do that. So in our first step, weβll
substitute π in place of ππΆ one.

In our next step, weβll want to
rewrite these combinations as permutations. We can do that by recognizing that
ππΆπ equals πππ over π factorial, which means we have π times six ππ six
over six factorial divided by seven ππ seven over seven factorial. And then in place of that division,
weβll multiply by the reciprocal such that we have π times six ππ six over six
factorial times seven factorial over seven ππ seven. We have six factorial in the
denominator and seven factorial in the numerator. Recall that we can rewrite seven
factorial as seven times six factorial. This means seven factorial over six
factorial simplifies to seven. From there, we have seven π times
six ππ six over seven ππ seven.

Notice that none of the four answer
choices have a seven π term, which means we need to think of some way to simplify
this further. To do that, we notice that we have
a seven π term and an π equals seven π. A property of permutations tells us
that πππ is equal to π times π minus one ππ minus one. That means seven ππ seven equals
seven π times seven π minus one π seven minus one. And in place of that seven minus
one, weβll have six. In our denominator, in place of
seven ππ seven, we can substitute the term seven π times seven π minus one π
six. And then the seven π in the
numerator and the denominator cancel, which gives us a simplified form of six ππ
six over seven π minus one π six, which is option (B).