Video Transcript
Which of the following is equal to
𝑛𝐶 one times six 𝑛𝐶 six over seven 𝑛𝐶 seven? (A) Six 𝑛𝑃 six over seven 𝑛
minus seven 𝑃 seven. (B) Six 𝑛𝑃 six over seven 𝑛
minus one 𝑃 six. (C) Six 𝑛𝑃 six over seven 𝑛
minus seven 𝑃 six. (D) Six 𝑛𝑃 six over seven 𝑛
minus one 𝑃 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 𝑛𝐶 one,
and the properties of combinations tell us that 𝑛𝐶 one just equals 𝑛. To choose one item from a set of
𝑛, there are 𝑛 different ways to do that. So in our first step, we’ll
substitute 𝑛 in place of 𝑛𝐶 one.
In our next step, we’ll want to
rewrite these combinations as permutations. We can do that by recognizing that
𝑛𝐶𝑟 equals 𝑛𝑃𝑟 over 𝑟 factorial, which means we have 𝑛 times six 𝑛𝑃 six
over six factorial divided by seven 𝑛𝑃 seven over seven factorial. And then in place of that division,
we’ll multiply by the reciprocal such that we have 𝑛 times six 𝑛𝑃 six over six
factorial times seven factorial over seven 𝑛𝑃 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 𝑛 times
six 𝑛𝑃 six over seven 𝑛𝑃 seven.
Notice that none of the four answer
choices have a seven 𝑛 term, which means we need to think of some way to simplify
this further. To do that, we notice that we have
a seven 𝑛 term and an 𝑛 equals seven 𝑛. A property of permutations tells us
that 𝑛𝑃𝑟 is equal to 𝑛 times 𝑛 minus one 𝑃𝑟 minus one. That means seven 𝑛𝑃 seven equals
seven 𝑛 times seven 𝑛 minus one 𝑃 seven minus one. And in place of that seven minus
one, we’ll have six. In our denominator, in place of
seven 𝑛𝑃 seven, we can substitute the term seven 𝑛 times seven 𝑛 minus one 𝑃
six. And then the seven 𝑛 in the
numerator and the denominator cancel, which gives us a simplified form of six 𝑛𝑃
six over seven 𝑛 minus one 𝑃 six, which is option (B).
