What Are All of the Possible 4-Digit Combinations for a Phone With 0 – 9 With Repeat?

Daniel Oines/CC-BY-2.0

There are 10,000 possible 4-digit combinations for a phone that dials 0 through 9 if repetition is allowed. Some of the combinations include {0,0,0,0}, {9,9,9,9} and all arrangements between. Since the order of digits matters ({0,0,0,1} is not equal to {0,0,1,0}) and repetition is allowed (each digit can be in any of the four positions), the arrangements are called “permutations with repetition.”

According to Math Is Fun, permutations with repetition are the easiest to calculate. The person needs to determine the number of dial pad numbers she can choose from (n) and the number of selections (r). In this case, to calculate a 4-digit number with numbers 0 through 9 available, there are 10 available dial pad numbers (n=10) for each digit with four digits selected (r=4). The number of permutations can be calculated with the formula n^r, in this case by multiplying 10 x 10 x 10 x 10.

When repetition is not allowed, the permutation formula is adjusted to the factorial function, which is noted by ?!? and is simply a multiplication of a series of descending natural numbers. The formula is useful in the cases when each number can be selected only once. For example, if there are 10 competitors in the race (n=10), and the first four get a reward (r=4), a person can calculate the number of possible reward distribution combinations by dividing n!, which is 10x9x8x7x6x5x4x3x2x1, with the factorial function of the difference between n and r (n-r)!.

When the order does not matter, such as in lottery drawings, the calculations are called combinations.