Question Video: Finding the Number of Ways to Arrange a Given Set of Digits and Form a 𝑛-Digit Number with Given Criteria | Nagwa Question Video: Finding the Number of Ways to Arrange a Given Set of Digits and Form a 𝑛-Digit Number with Given Criteria | Nagwa

Question Video: Finding the Number of Ways to Arrange a Given Set of Digits and Form a 𝑛-Digit Number with Given Criteria Mathematics • Third Year of Secondary School

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How many four-digit numbers, with no repeated digits, can be formed using the elements of the set {0, 1, 3, 4}?

02:15

Video Transcript

How many four-digit numbers with no repeated digits can be formed using the elements of the set containing the numbers zero, one, three, and four?

Now, what we don’t really want to do is use systematic listing. We don’t want to try listing out all possible four-digit numbers, for two reasons. Firstly, there might be quite a lot of them. And secondly, it would be really easy to miss some out. Instead, we’re going to use something called the product rule for counting. This says that we can find the total number of outcomes when combining two or more events by multiplying the number of outcomes of each event together. We say it’s called the product rule for counting because it involves finding a product by multiplying.

So, let’s look at what we’re trying to find. We want a four-digit number with no repeated digits. And we’re going to use the set containing zero, one, three, and four. So, let’s choose our first digit of our four-digit number. It follows that since it’s a four-digit number, the first digit can’t be zero. If it was, we would technically have a three-digit number, which means the first digit in our number can either be one, three, or four. There are then three ways of choosing the first digit in our four-digit number.

Let’s now consider the second digit. We’ve already chosen an element from our set. This means there are now only three numbers left in the set. There are three ways of choosing the second digit. Remember, we want no repeated digits, so we have to disregard the first digit we used. We’ll now consider the third digit in our number. At this point, we’ve chosen two elements of our set. We don’t know what elements we’ve chosen, but we do know we can’t repeat them.

So, there are two elements left to choose from for our third digit. It follows, of course, that there is only one way or one element we can choose for our fourth digit. The product rule says that to find the total number of outcomes, we multiply the number of outcomes of each event together. So, the total number of four-digit numbers we can make is found by multiplying three by three by two by one, which is 18. We could make 18 four-digit numbers with no repeated digits using the elements zero, one, three, and four.

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