Video Transcript
Mia’s teacher divided the class
into groups of six and required each member of the group to meet every other member
of the same group. How many meetings will each group
have?
In this question, we are told that
Mia’s class has been divided into groups of six students. We are told that every member of
each group of six must meet with all of the other members. We need to determine how many
meetings will take place inside each group of six. There are two different ways we can
answer this question. We can start by noting that a
meeting in this context means we choose any two students from the same group, that
is, the two members of each meeting. In particular, we can note that the
order of the two students we choose does not matter. For instance, a meeting between Mia
and Emma will be the same as a meeting between Emma and Mia.
We can then recall that 𝑛 choose
𝑘 is the number of ways of choosing 𝑘 objects from 𝑛 distinct objects, provided
𝑛 is greater than or equal to 𝑘 and they are both natural numbers. If we substitute 𝑛 equals six and
𝑘 equals two into this expression, then we see that six choose two is the number of
ways of choosing two objects from six objects. In this case, it will give us the
number of ways of choosing two students from any group of six. Since each choice of two students
from the six gives us a meeting, this is the number of meetings that each group will
need to have.
We can calculate this value by
recalling that 𝑛 choose 𝑘 is equal to 𝑛 factorial over 𝑘 factorial times 𝑛
minus 𝑘 factorial. We can substitute 𝑛 equals six and
𝑘 equals two into the formula to obtain six factorial over two factorial times six
minus two factorial, which simplifies to give four factorial. We can then evaluate this
expression by recalling that a factorial of 𝑛 is the product of all positive
integers less than or equal to 𝑛. So, six factorial is six times five
times four factorial. We can then cancel the shared
factor of four factorial in the numerator and denominator to get six times five over
two factorial. We can then calculate that two
factorial is equal to two. So this expression evaluates to
give 15.
It is worth noting that we can
answer this question without using combinations at all. First, we can note that each group
has six members. To calculate the total number of
meetings this group must make, we can note that one student will meet five other
members of the group. For the next student, we have
already considered their meeting with the first student. So the student must have four more
meetings, one with each of the other members. For the third student, we have
already considered their meeting with the first two students. So they must make three more
meetings, one with each of the other students.
We can keep applying this
process. The fourth student will have two
more meetings to attend. And the fifth student will have one
more meeting. The sixth student will have already
met with all five of the other members at this point. The sum of these values is equal to
15. This also highlights a link between
combinations and sums. We can see that 𝑛 choose two is
equal to the sum of the positive integers below 𝑛. In any case, we have shown that the
total number of meetings required in a group will be 15.
