Video Transcript
We have four number cards that are numbered three, four, five, and six. How many unique numbers can we form using at least one of the cards?
Reading the question carefully, we’re asked how many unique numbers we can form using at least one of the four cards. This means that we could use just one card to form a single-digit number, two cards to form a two-digit number. We could use three cards to form a three-digit number. Or we could use all four to form a four-digit number. We’ll need to consider each of these possibilities separately.
If we’re just using one card, then fairly logically we can see that there are four possibilities. We could use each of the cards, either three, four, five, and six. So, there are four unique numbers that can be formed if we use just one of the cards. Let’s think about if we’re using two cards and we need to be a little bit more careful here. We can do this by considering how many options there are for each digit in the two-digit number.
For the first digit, there are four possibilities, either three, four, five, or six. But when we get to the second digit, we’ve already placed one of the cards down to form the first digit. So, there are only three possibilities left. Any one of the first digits can be combined with any of the second digits. So, we have four multiplied by three, which is 12, possible two-digit numbers.
Now, it’s important to note that the order matters here. The number 56 is different to the number 65, so we’re counting this pair of cards twice in its two possible orderings. What we’re actually looking at here are permutations, the number of unique ways we can select two cards from four when the order matters. This can be written using the notation four 𝑃 two. In general, the notation 𝑛𝑃𝑟 means the number of distinct permutations of 𝑟 objects from 𝑛 distinct objects. And this is calculated using the formula 𝑛 factorial over 𝑛 minus 𝑟 factorial.
You may also see factorials denoted using an exclamation mark. Returning then to the number of options when we use two cards, we could have worked this out by evaluating four 𝑃 two, which is four factorial over four minus two. That’s two factorial. Four factorial is four times three times two times one; that’s 24, and two factorial is two times one, which is two. So, we have 24 over two which is indeed equal to 12.
When we come to the number of distinct numbers we can form using three cards then, we can go straight to using a permutation. The number of ways we can select three cards from four when the order matters is four 𝑃 three, which is four factorial over four minus three or one factorial. One factorial is equal to one and four factorial as we’ve already seen as equal to 24. So, we have 24 over one, which is 24.
We can confirm this by considering the number of options for each card. There are four options for the first card, then three for the second card because we’ve just used one, and then finally two for the third card because we’ve already used two. The total number of possible orderings of three cards from four then is four multiplied by three multiplied by two, that’s 12 multiplied by two, which is indeed 24.
Finally, the number of unique numbers we can form using all four cards is four 𝑃 four, which is four factorial over four minus four or zero factorial. At this point, we have to recall that zero factorial is defined to be equal to one. So, we have 24 over one, which is equal to 24. Considering again the number of options for each card, we have four multiplied by three multiplied by two multiplied by one, which is 24.
So, we’ve worked out the number of unique numbers that can be formed using one card, two cards, three cards, and finally all four cards. To work out the total number of unique numbers that can be formed, we need to add these together: four plus 12 plus 24 plus 24, which is 64. The number of unique numbers that can be formed using at least one of the four distinct number cards is 64.
