Who is asking: Jessie
Grade 11 student

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?

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 10⁸¹, that is 8 followed by 81 zeros.

The number of atoms in the universe is somewhere between 10⁶⁹ and 10⁸¹ 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 10¹⁰⁰⁰. The exact number is

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 10¹⁸ 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 6-49 lottery every Wednesday and Saturday it would take you, on the average, 70,000 years to win.