# 1. How many different ways can you have r numbers 1 sum up to a number...

1. How many different ways can you have r numbers 1 sum up to a number n? These are called compositions of a number n and it is easy to calculate from our understanding of binomial coefficients. So the number of compositions of 4 into 3 parts will be 1+1+2, 1+2+1, and 2+1+1. Note how we think of 1+1+2 and 1+2+1 as different-because in the first case, the first number is 1, second is 1 and third is 2, while in the second case, the first is still 1 but the second is 2 and the third is 1 (so a different choice). 2. A seemingly related problem is where you break up a number n into parts, but do not distinguish between how you arrange the parts. This is called the number of partitions of a number.
A second definition for partitions, one I prefer that avoids confusion, is to define the number of partitions of n as the number of ways of writing n as a sum of numbers all arranged in ascending order. Explain why the two definitions above give the same value to the partition. So the only way to partition 4 into 3 parts is to write it as 1+1+2. All partitions of 4 are 4, 1+3, 2+2, 1+1+2, and 1+1+1+1- so there are 5 of them. While they "appear" related to compositions, they could not be more different. Counting compositions is something I can assign in an elementary undergraduate class. Partitions are notoriously difficult to compute (but there is a quick and elegant way to approzimate them). 3. The number of partitions grows roughly as exp The story of how we computed it is quite remarkable, it is a story that unfolded over more than a century, starting all the way from Euler and ending with Hardy and Ramanujan. Now how many different compositions of n are there? 4. Another related problem is the number of ways you could write n as a sum of r numbers 2 0. You can easily look this result up online, but I do not want you to do that. Instead derive this number by using part 1. above.

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