There are 10,000 different ways to use four numbers when each number is used more than once. There are also 5,040 ways to put four numbers together when each number is used only once.
How so? Well, for each number in the combination, you can choose from 0 through 9. Since there are four numbers in the combination, there are 10 different ways to choose each of the four numbers. That is, there are 10,000 possible combinations, which is equal to 104 or 10,000.
The binomial coefficient formula is a general way to figure out how many different ways there are to do something. Here, the number of ways that k items from a set of n items can be put together is n!/(k!*(n-k)!, where the exclamation point means a factorial. Need to go into more detail? We’ll take care of you.
Formula for the Number of Combinations
A simple equation can be used to find out how many different ways four numbers can be put together. Think of each number as a person and each spot in the combination as a seat. There is only room for one person in each seat, and there are a total of 10 seats. (Because single-digit numbers go from 0 to 9, there are 10 numbers.)
In any given combination, any one of the 10 numbers can go in any of the four spots. For the first seat, there are 10 different ways to choose. Also, for the second seat, there are 10 options that can be put together in any way. The same is true for seats three and four. To find the total number of options for all possible combinations, multiply the number of options for the first seat by the number of options for the second seat by the number of options for the third seat by the number of options for the fourth seat.
That is, you need to multiply 10 by 10 by 10 by 10 by 10. In the end, you’ll find that there are 10,000 different ways that four numbers can be put together.
Numbers that don’t repeat have the same value only once.
You would be right and wrong if you said that there are 10,000 possible ways to put four numbers together. That is, the answer of 10,000 makes it possible for any of the 10 numbers to sit in any of the four seats. Based on this theory, one of the 10,000 possible combinations could be 1111, 0000, 2222, or 3333. Let’s add something new to the mix.
In the real world, most four-digit combinations don’t have numbers that come back again and again. In fact, many companies don’t let people set up four-digit passwords that repeat the same number over and over. So, how many ways are there to make a four-digit number where the numbers don’t repeat?
Forget about the seats for a moment and use the binomial coefficient formula, which is a very useful piece of math. Here’s how to figure it out:
n!/(k! x (n-k)!)
If you didn’t already know, each exclamation point stands for a factorial. Even though both the name and the formula make it sound hard to do, it’s actually much simpler. It turns out that the idea of people sitting down will also help with this one. “K” is the number of people who can sit in any one of the seats, and “n” is the number of seats any of those people can sit in.
If you want to know how many ways you can put four numbers together, k=10 and n=4. This is how the equation looks:
4!/(10! x (4-10)!)
Without getting into the details, this means:
10 x 9 x 8 x 7 = 5,040
Notice a pattern here? Any of the 10 numbers can sit in the first seat. Now, there are only nine numbers left that can sit in the second seat. With one more number gone, there are only eight numbers that could sit in the third seat, and there are only seven numbers that could sit in the fourth seat.
See? The binomial coefficient is much easier to understand than it might seem. Using the binomial coefficient, if a number is picked for one seat, it is no longer a possibility for the other seats. This cuts the number of possible combinations by about half.
What This Says About the Password for Your Smartphone
Let’s be honest. Unless you’re really, really into numbers, you probably didn’t look up how many ways four digits can be put together. In reality, you probably ended up here because you’re trying to set a four-digit password. And it’s great that you’re giving your passcode a lot of thought.
Since four-digit passwords are some of the shortest you’re likely to use, they might seem easy. But they are also often among the most important. You can use four-digit number combinations to open your phone or sign in to some apps faster, but where else can you use them? Most banks ask customers to choose a four-digit PIN so that they can use ATMs and authorise transactions.
Hackers take advantage of the fact that four-digit number combinations are used as passwords for things you probably care less about protecting than your bank card PIN. When it comes to passwords, people don’t come up with nearly as many good ideas as they should. If someone can figure out your lock screen code, it’s likely that they can also use your debit card to make a purchase. After all, there’s a very good chance that the numbers on your debit card and your lock screen code will be the same.
Even banks don’t help solve the problem. People often have 10,000 options for PINs because many banks let people use the same number more than once. If your bank is a little smarter about security, you’ll only have 5,040 options. A lot of people use four-digit combinations that repeat or are in order. For example, 1234 is a very popular choice, and some people use the same number over and over, like 1111 or 2222.
Don’t just know about the binomial coefficient and do nothing with it. You could choose from thousands of different ways to put together four numbers. Do not just choose your year or date of birth. Please don’t pick 1234, for the love of all that is good. If you don’t want a certain person to look at your phone, you’ll have to work much harder than that. Choose your passwords wisely to protect your information and identity.
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