# Question Video: Determining Equivalent Expressions Involving Combinations and Permutations Mathematics

Which of the following is equal to (ππΆβ Γ 6ππΆβ)/7ππΆβ? [A] 6ππβ/(7π β 7)πβ [B] 6ππβ/(7π β 1)πβ [C] 6ππβ/(7π β 7)πβ [D] 6ππβ/(7π β 1)πβ

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### Video Transcript

Which of the following is equal to ππΆ one times six ππΆ six over seven ππΆ seven? (A) Six ππ six over seven π minus seven π seven. (B) Six ππ six over seven π minus one π six. (C) Six ππ six over seven π minus seven π six. (D) Six ππ 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 ππΆ one, and the properties of combinations tell us that ππΆ 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, weβll substitute π in place of ππΆ one.

In our next step, weβll want to rewrite these combinations as permutations. We can do that by recognizing that ππΆπ equals πππ over π factorial, which means we have π times six ππ six over six factorial divided by seven ππ seven over seven factorial. And then in place of that division, weβll multiply by the reciprocal such that we have π times six ππ six over six factorial times seven factorial over seven ππ 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 ππ six over seven ππ 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 πππ is equal to π times π minus one ππ minus one. That means seven ππ seven equals seven π times seven π minus one π seven minus one. And in place of that seven minus one, weβll have six. In our denominator, in place of seven ππ 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 ππ six over seven π minus one π six, which is option (B).