There are 10,000 combinations of four numbers when numbers are used multiple times in a combination. There are 5,040 combinations of four numbers when numbers are used only once.
There are 10 choices, zero through nine, for each number in the combination. Because there are four numbers in the combination, the total number of possible combinations is 10 choices for each of the four numbers. That is, the number of possible combinations is 10*10*10*10 or 10^4, which is equal to 10,000. The binomial coefficient formula is a general way to calculate the number of combinations. The number of combinations of k elements from a set with n elements is n!/(k!*(n-k)!), in which the exclamation mark indicates a factorial.
To calculate the number of combinations in which each digit is used only once, recognize that the number of digit choices is different for each slot in the combination. There are 10 choices for the first slot, nine for the second, eight for the third and seven for the fourth. The number of possible combinations in which digits do not repeat is 10*9*8*7 or 5,040. In general, the number of combinations of k elements from an n-element set is n!/(n-k)! when each element appears only once in a combination.