Who is asking: Jessie My teacher told me that the number of ways of lining up the 450 students in our school is larger than the number of atoms in the universe. Is this right? Hi Jessie, Think first about lining up a small number of students. If you have two students, John and Mary you can line them up in 2 ways, John and Mary or Mary and John. If there are three children then you can line them up in 6 ways. There are 3 choices for the first position, 2 for the second and then 1 for the remaining child. Hence there are 3 x 2 x 1 = 6 possible arangements. If you have 60 children in your classroom then the number of ways you can line them up is 60 x 59 x 58 x ... x 3 x 2 x 1 ways. This number is approximately 8 x 10^{81}, that is 8 followed by 81 zeros. The number of atoms in the universe is somewhere between 10^{69} and 10^{81} so, even with only 60 students, the number of ways you can line them up is larger than the number of atoms in the universe. 60 x 59 x 58 x ... x 3 x 2 x 1 is called 60 factorial. Factorials are clearly enormous numbers. If you have 450 children you can line them up in 450 factorial ways. This is approximately 1.7 x 10^{1000}. The exact number is 51015569207509555333001683436750604675088290438710 61458112845184240978586185838063016502083472961813 51667570171918700422280962237272230663528084038062 31236934267413503661010150883822049497092973901163 67937661650237308538964039015908361441495944326842 04513784716402303182604094683993315061302563918385 30334151060676146242020582000693635209596741718319 15387256175095213805567813091954298002292738033425 53558164591996298912368598547771179158461351340068 90564712765816483637712630377492336007807230746200 85543550683614481266062811457609604991878134283979 24840592504537849487425060488481036571447957046788 63574293671461517621914846974310297994974071448510 47161696640523973926028484086940074089989011274929 05171514473431386633392492040661522692303043813960 54196609322424380922513726885171790430321405823844 79361116785682369730362384046265078906880000000000 00000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000 Just to get a sense of how large these numbers are, suppose that you have just 20 students and they are very fast. Every second they can arrange themselves in a different one of the 20 factorial possible lineups. 20 factorial is approximately 2.4 x 10^{18} and the number of seconds in a year is 31,536,000 so it would take them approximately years to go through each of the 20 factorial possibilities. This is about 77 billion years. We don't have a very good sense about large numbers. For example if you play the 649 lottery every Wednesday and Saturday it would take you, on the average, 70,000 years to win. Cheers,Denis and Harley
