Video Transcript
If nine choose 𝑟 is equal to nine
choose three, find all the possible values of 𝑟.
Let’s begin simply by recalling
what we understand by 𝑛 choose 𝑟. 𝑛 choose 𝑟 or 𝑛𝐶𝑟, which
represents the number of combinations of size 𝑟 taken from a collection of 𝑛
items, is defined as 𝑛 factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. We can, therefore, say that the
left-hand side of our equation nine choose 𝑟 can be written as nine factorial over
𝑟 factorial times nine minus 𝑟 factorial. We can then write the right-hand
side nine choose three as nine factorial over three factorial times nine minus three
factorial or simply nine factorial over three factorial times six factorial.
Now, of course, these are equal to
one another. In other words, nine factorial over
𝑟 factorial times nine minus 𝑟 factorial is equal to nine factorial over three
factorial times six factorial. So by comparing both sides, we
could relate 𝑟 to this value here. In this case, we’re saying that 𝑛
is equal to nine and 𝑟 is equal to three. But, in fact, there is another
value of 𝑟. This value of 𝑟 comes from the
fact that multiplication is commutative; it can be performed in any order. In other words, we can reverse the
terms on our denominator. That is, we can reverse the three
and the six.
In doing so, we don’t actually
change the value of our expression. But we can now say, “Well, 𝑛 must
be equal to nine, and 𝑟 must be equal to six.” If this is the case, then nine
minus 𝑟 should then be equal to three. Well, letting 𝑟 be equal to six,
we see that nine minus 𝑟 is nine minus six, which is indeed equal to three. And so there’s a second value of 𝑟
we could choose. 𝑟 could be equal to six. If nine choose 𝑟 is equal to nine
choose three, we can say 𝑟 could be equal to three or 𝑟 could be equal to six.
