### Video Transcript

In how many ways can an odd number
of six digits be formed using the numbers one, two, three, four, five, six if no
digits are to be repeated?

To answer this question, we’re
going to use something called the product rule for counting. The product rule for counting can
save us a little bit of time because we don’t need to list out every possible number
we can make. And it says to find the total
number of outcomes for two or more events, we multiply the number of outcomes for
each event together. Here we have six digits. And we’re looking to make an odd
number. Well, an odd number will end in the
digits one, three, or five. And so we’ll just begin by working
out how many ways there are for us to make an odd number.

Let’s imagine we’re choosing the
final digit of our six-digit number from this list. We begin by picking that final
digit, and so there are three ways to choose that digit. It can be a one, a three, or a
five. Since we’ve already picked a digit,
we know that the next digit, the second digit in our six-digit number, will be
chosen out of the remaining five. Then there are only four digits
left. So there are four ways to choose
the third digit. We then choose the fourth
digit.

Well, there are only three numbers
left to choose from. So there are three ways to choose
this number. There are two ways to choose the
fifth number. And then we have no choice on the
sixth number. There is just one way to choose
that. And so the number of ways of
choosing an odd number of six digits from our list, assuming that none are repeated,
is three times five times four times three times two times one, which is equal to
360. There are 360 ways to choose an odd
number using our six digits.