I need to know all the 4 digit codes you can make with 1234567890

Expert Answers
astrocourt eNotes educator| Certified Educator

There is a mathematical formula which will produce the number of combinations, or permutations, from a number of selections.

Note that combinations and permutations have different mathematical meanings. Permutations have a specific order: i.e. 123 is different to 321. However, for combinations, the order does not matter. That is, 123 and 321 are the same combination.

You also need to consider whether replacement is allowed. For example, is 4444 a code allowed by your selection - or can there only be one instance of a 4?

The number of possible permutations with replacement is given by the formula `n^r` where you have n choices and your code is r long. In your example, you have 10 digits and you want a 4 digit code. So there will be `10^4=10000` different codes.

If we want the number of permutations without replacement, the formula is given by: `(n!)/((n-r)!)`  . So in this case:   `(10!)/((10-4)!) = 3628800/(6!) = 3628800/720 = 5040`  different codes.

If the order doesn't matter, we can use combinations instead of permutations. 

The number of combinations with replacement is given by `(n!)/((r!(n-r)!))`

in this case the formula gives 210 combinations.

The number of combinations without replacement is given by `((n+r-1)!)/((r!(n-1)!))`

in this case, the formula gives 715 combinations.

` `