Video Transcript
How many ways can a committee of
one man and one woman be formed from 83 men and 12 women?
To answer this question, we’re
going to recall something called the fundamental counting principle or the product
rule for counting. This tells us that the total number
of outcomes for two or more combined events is found by multiplying the total number
of outcomes for each event together. And so we next need to ask
ourselves, what are the events we’re interested in? Well, we’re forming a committee of
one man and one woman. So event number one will be
choosing that man. And then it follows that event
number two is choosing one woman. In fact, we’re choosing one out of
a possible 83 men. And so there are 83 outcomes for
event one.
Similarly, we’re choosing one out
of a possible 12 women. And that means that there are 12
possible outcomes for event two. We can choose from 12 possible
women. The product rule for counting or
the fundamental counting principle tells us that the total number of outcomes, which
is the number of ways a committee of one man and one woman can be formed, is the
product of these two numbers. It’s 83 times 12. And of course, we might look to
perform that on a calculator. But in the absence of one, there
are a number of written methods we can use. Let’s recall the column method.
We begin by multiplying three by
two. Three times two is six. Then we multiply the eight on our
top number by that two again. Now eight times two is 16, so this
becomes 166. Our next job is to multiply each of
the digits in the number 83 by this one. But of course, this is really
10. And so we’re actually going to add
a zero in this column to show that we’re multiplying by 10 rather than just one. Three times one is three, and eight
times one is eight.
Our final step is to find the sum
of these two three-digit values. Six plus zero is six, six plus
three is nine, and one plus eight is nine. And so 83 times 12 is 996. And so we see there are 996 ways to
form a committee of one man and one woman from a total of 83 men and 12 women.
