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In how many ways can a four-digit even number with no repeated digits be formed using the elements of the set containing the numbers one, six, nine, eight, seven, five?
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To calculate the number of ways that we can create a four-digit even number, we don’t really want to use systematic listing.
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We don’t want to list all the possible options.
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Firstly, there could be quite a lot of them.
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And secondly, it’s really easy to lose some.
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And so we recall the product rule for counting.
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This says that we can find the total number of outcomes for two or more events by multiplying the number of outcomes for each event together.
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In this case, our events are the digits we choose.
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Now, we actually have six elements in our set.
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We’re only interested in calculating the number of four-digit numbers we can make.
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And so let’s just begin by considering what we mean for it to be an even number.
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Well, from our set, we know that the even numbers will all end in a six or an eight.
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So we’ll consider the last digit of our four-digit number first.
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And we say that, well, if the even numbers must end in a six or an eight, there are only two ways to choose the last digit of our even number.
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It can be six or eight.
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There are no other restrictions on our four-digit number.
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And so once we’ve chosen the last digit, we’ve either chosen a six or an eight, we know that there are only five ways to choose the next digit.
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Since there are no repeated digits, this means that there are four ways to find the next digit in our number.
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And finally, once we’ve chosen the first three digits, we know that there are three elements left in our set.
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So there are three ways to choose the fourth digit.
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The product rule says to find the total number of outcomes, we multiply the number of outcomes of each event together.
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So here that’s two times five times four times three.
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Well, two times five is 10, and four times three is 12.
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So we can calculate this quite quickly by working out 10 times 12, which is 120.
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This means there are 120 ways to choose a four-digit even number with no repeated digits from our set.