How Many Different Combinations Are Possible Using Four Numbers?

When a number is used more than once in a combination, there are 10,000 possible combinations of four numbers. When a number is used just once, there are 5,040 possible combinations of four numbers.

How so? For each combination of numbers, there are ten options, numbered from 0 to 9. There are a total of 10 options for each of the four integers because there are four numbers in the combination. In other words, there are 10,000 different combinations out of 101010*10 or 104.

The number of combinations can be determined generally using the binomial coefficient formula. The number of ways to combine k items from a set of n elements is shown here as n!/(k!*(n-k)!), where the exclamation point denotes a factorial. Do we need to elaborate more? We have your back.

Formula for Number of Combinations

There is a straightforward equation that may be used to determine how many possible possibilities there are for four numbers. Consider each digit in the combination as a person, and each position as a seat. There can only be one person in each seat, and each seat can only hold ten people. Because single-digit numbers range from 0 to 9, there are ten numbers.

Any one of the 10 numbers may occupy one of the four seats in any particular combination. There are ten possible configurations for the first seat. Additionally, there are 10 possible combinations for the second seat.

The third and fourth chairs also fit this description. 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 to get the total options for all possible combinations.

Therefore, you must multiply 10 by 10 by 10 by 10. There are 10,000 possible combinations of four numbers, you’ll discover in the end.

Formulas for Combinations of Numbers When They Don’t Repeat

You would be correct and incorrect if you said that there were 10,000 potential combinations using just four integers. In other words, the 10,000 solution allows any one of the 10 numbers to occupy any one of the four seats.

According to this hypothesis, one of the 10,000 combinations may be one of the following: 1111, 0000, 2222, or 3333. Let’s introduce a complication to the situation.

Four-digit combinations frequently lack repeated digits in the real world. In fact, a lot of businesses forbid users from creating four-digit passwords that contain the same number repeatedly. So how many non-repeating four-digit number combinations are there in total?

For a moment, ignore the chairs and focus on the binomial coefficient formula, a handy-dandy mathematical formula. The equation reads as follows:

n!/(k! x (n-k)!)

The factorial is represented by each exclamation point, in case you didn’t know. Although the term and the formula both appear difficult, they are actually considerably simpler in use. It turns out that the idea of people sitting in seats will be useful for this one as well. The number of individuals who can fit in any one of the seats is denoted by “K,” and the number of seats that any one of those persons can take is denoted by “n.”

When trying to determine how many possible combinations there are of four numbers, k=10 and n=4. This is how the equation looks:

4!/(10! x (4-10)!)

That translates to: without factorization

10 x 9 x 8 x 7 = 5,040

Do you see a pattern here? Any one of the 10 numbers may take the first seat. There are now just nine numbers available for the second seat. The third seat can only hold eight additional people after the final number is down, while the fourth seat can only hold seven numbers in total.

See? Contrary to appearances, the binomial coefficient is much easier. The binomial coefficient ensures that any number selected for one seat is automatically discarded from consideration for the other seats. This roughly reduces the number of possible combinations.

Observations on Your Smartphone Password

Let’s be real here. You probably didn’t look up the amount of possible combinations of four digits unless you have a serious interest in numbers. Actually, the reason you’ve probably arrived at this section of the internet is because you’re attempting to create a four-digit password. And it’s admirable that you’re considering your passcode.

Given that they are among the smallest passwords you’re likely to use, four-digit passwords can appear to be quite straightforward. They also frequently rank among the most significant, though.

Four-digit number combinations can be used to open your phone or log in to some apps more quickly, but how else do you use them? To authorise transactions and utilise ATMs, the majority of banks require their customers to choose a PIN with four digits.

Four-digit number combinations are frequently used as passwords for items you probably care less about securing than your bank card PIN, which gives hackers an opening. When it comes to passwords, people are not nearly as creative as they should be.

Since there is a very high likelihood that the numbers will match, if someone can guess the code on your lock screen, it is possible that they can also authorise a transaction on your debit card.

Banks also contribute little to the issue. People frequently have 10,000 options for PINs because many banks permit repeating numbers. You’ll only have 5,040 choices to pick from if your bank is a little more security-conscious.

Many people employ four-digit sequences that are repetitive or sequential. For instance, many people choose 1234, whereas some people repeatedly combine the same number, such as 1111 or 2222.

Don’t waste your understanding of the binomial coefficient. There are literally tens of thousands of possible four-number combinations. Do not simply choose your birth year or day. Please, for the love of everything good, don’t choose 1234 either.

You’ll need to work much more than that to prevent a certain someone from prying eyes your smartphone. Be careful when selecting passwords to protect your identity and personal data.


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