different. According to the pigeonhole principle, is zero when . When : where is the factorial operator, is the binomial coefficient and denotes permutation. The equation expresses the fact that the first person has no one to share a birthday, the second person cannot have the same birthday as the first (), the third cannot have the same birthday as either of the first two (), and in general the th birthday cannot be the same as any of the preceding birthdays. The event of at least two of the persons having the same birthday is complementary to all birthdays being different. Therefore, its probability is The following table shows the probability for some other values of (for this table, the existence of leap years is ignored, and each birthday is assumed to be equally likely): {| class="wikitable" !!! |- |align=right|1 || 0.0% |- |align=right|5 || 2.7% |- |align=right|10 || 11.7% |- |align=right|20 || 41.1% |- |align=right|23 || 50.7% |- |align=right|30 || 70.6% |- |align=right|40 || 89.1% |- |align=right|50 || 97.0% |- |align=right|60 ||99.4% |- |align=right|70 || 99.9% |- |align=right|75 || 99.97% |- |align=right|100 || % |- |align=right|200 || % |- |align=right|300 || (100 − )% |- |align=right|350 || (100 − )% |- |align=right|365 || (100 − )% |- |align=right|≥ 366 || 100% |} Approximations The Taylor series expansion of the exponential function (the constant ) provides a first-order approximation for for : To apply this approximation to the first expression derived for , set . Thus, Then, replace with non-negative integers for each term in the formula of until , for example, when , The first expression derived for can be approximated as Therefore, An even coarser approximation is given by which, as the graph illustrates, is still fairly accurate. According to the approximation, the same approach can be applied to any number of "people" and "days". If rather than 365 days there are , if there are persons, and if , then using the same approach as above we achieve the result that if is the probability that at least two out of people share the same birthday from a set of available days, then: Simple exponentiation The probability of any two people not having the same birthday is . In a room containing n people, there are pairs of people, i.e. events. The probability of no two people sharing the same birthday can be approximated by assuming