Video Transcript
A teacher must pick a committee of
six students consisting of four boys and two girls. In how many ways can the committee
be formed if the teacher has eight boys and seven girls to choose from?
In this question, we are told that
a teacher needs to select a committee consisting of six total students, four boys
and two girls. We want to determine the total
number of possible committees if the teacher has eight boys and seven girls to
choose from. To do this, we can start by
recalling that 𝑛 choose 𝑘 is the number of ways of choosing 𝑘 objects from 𝑛
distinct objects. We can use this to find the number
of ways of choosing 𝑘 things from 𝑛 things, provided the order we choose does not
matter.
In this case, we know that the
order we choose the committee does not matter. The only differences are who is
chosen for the committee. We can also recall that 𝑛 choose
𝑘 is equal to 𝑛 factorial over 𝑘 factorial times 𝑛 minus 𝑘 factorial. Let’s use this result to determine
the number of ways of choosing four boys from the eight candidates. We have 𝑛 equals eight and 𝑘
equals four, so this is given by eight choose four. Substituting these values into the
formula yields eight factorial over four factorial times eight minus four
factorial. We can simplify this expression to
obtain eight factorial over four factorial times four factorial.
We can simplify further by
recalling that the factorial of a number is the product of the positive integers
less than or equal to the number. In particular, eight factorial can
be written as eight times seven times six times five times four factorial. Canceling the shared factor of four
factorial in the numerator and the denominator and then evaluating gives us that
there are 70 ways of choosing four boys from the eight candidates who are boys.
We can follow the same process to
determine the number of ways of choosing two girls from the seven candidates who are
girls. We substitute 𝑛 equals seven and
𝑘 equals two into the formula to get seven factorial over two factorial times seven
minus two factorial. We can then simplify and evaluate
this expression in the same way. We have that seven factorial over
five factorial is equal to seven times six. Then we divide this by two
factorial to obtain 21. Hence, there are 21 ways of
choosing two girls from seven to be on the committee.
We now need to combine these
results to find the total number of ways the teacher can select the entire
committee. To do this, we first note that
choosing the boys and choosing the girls for the committee are independent. The choices of one group will not
affect the other. Since the events are independent,
we can use the fundamental counting principle to find the number of ways of choosing
the committee. We recall that this tells us that
we take the product of the number of ways of choosing from each set to find the
total number of possibilities. Hence, the number of different
committees that can be formed by the teacher is 70 times 21, which is equal to
1,470.
