In a gallery, each painting is referenced by two distinct English letters and a four-digit number which has no zeros and no repeated digits. How many paintings can be referenced using this system?
To answer this question, we’re going to begin by splitting it into two parts. Firstly, the reference number contains two distinct English letters. And so we’re going to be considering the 26 letters in the English alphabet. Then we have a four-digit number which contains some restrictions. We’re going to use something called the product rule for counting or the counting principle. This says that to find the total number of outcomes for two or more events, we multiply the number of outcomes of each event together.
So let’s begin by working out the total number of ways of ordering our English letters. We begin by considering the total number of ways of finding the first letter. Well, there are 26 letters in the alphabet, so there are 26 ways of finding the first letter. Once we’ve chosen that letter, we have only 25 left to choose from. We’re told that the letters are distinct. They’re different. And so the total number of ways of choosing our English letters is 26 times 25, which is 650.
The problem is in doing this, we end up with some duplicates. We could have AB, for example, or BA. These are actually the same pairing. And what that means is we’ve actually overcounted the number of ways of ordering our English letters. To take into account these doubles or these duplicates, we need to divide the number of ways of ordering our letters by two. That’s 650 divided by two, which is 325. And so we say that there are 325 ways of choosing our distinct English letters.
And what about the four-digit number? Well, it has no zeros and no repeated digits, meaning that we can use the numbers one through nine. And so we can say that the number of ways we can choose our first digit in our four-digit number must be nine. We can choose any of the numbers one through nine. However, because the number cannot be repeated, we now know that there are only eight ways of choosing the second digit. We’ve already chosen one of the numbers one through nine, so there are eight left to choose from. In the same way, once we get to the third digit, we’ve already chosen two. So there are only seven left to choose from. Finally, when we get to the fourth digit, we only have six left to choose from.
So the number of ways of choosing our four-digit number is nine times eight times seven times six, which is 3,024. Now we want to combine these sets of outcomes, the number of ways of ordering our letters and the number of ways of ordering our number. So we’re going to multiply the total number of outcomes for each event together. That’s 325 times 3,024. That’s 982,800. The total number of paintings that can be referenced using this system, then, is 982,800.