There are six ways to write 5 as an ordered sum of three positive integers: 3+1+1, 1+3+1, 1+1+3, 2+2+1, 2+1+2, 1+2+2.
A) What is the number of ways to write "n" as an ordered sum of "r" integers?
B) What is the number of ways to write "n" as an ordered sum of positive integers? For example, when n=3 the answer is four: 3, 1+2, 2+1, 1+1+1
A) To find the number of ways to write "n" as an ordered sum of "r" integers, we can use a mathematical concept called partitions. A partition of a number "n" into "r" positive integers represents the different ways to express "n" as a sum of "r" positive integers, taking the order into account.
One way to approach this problem is to use recursion. We can define a function P(n, r) that gives us the number of ways to write "n" as an ordered sum of "r" positive integers.
The base case is when r = 1, in which case there is only one way to write "n" as an ordered sum of one positive integer (simply "n" itself). So P(n, 1) = 1.
For the recursive case, we can consider the first integer in the sum. It can be any integer between 1 and "n". If we fix the first integer as k, then we need to find the number of ways to write ("n" - k) as an ordered sum of ("r" - 1) positive integers. We sum up these counts for all possible values of k to get the final result.
So, the recursive formula becomes:
P(n, r) = P(n-1, r-1) + P(n-2, r-1) + ... + P(n-r+1, r-1) + P(n-r, r)
Using this recursive formula, we can calculate P(n, r) for any given values of "n" and "r".
B) To find the number of ways to write "n" as an ordered sum of positive integers without fixing the number of integers, we can use a similar approach.
First, we find the number of ways to write "n" as an ordered sum of "r" integers, where "r" ranges from 1 to the maximum possible number of integers. We sum up these counts to get the total number of ways to express "n" as an ordered sum of positive integers.
Let's illustrate this with an example:
For n = 3, we calculate the following:
P(3, 1) = 1 (1 way to write 3 as an ordered sum of 1 integer: 3)
P(3, 2) = P(2, 1) + P(1, 1) = 1 + 1 = 2 (2 ways to write 3 as an ordered sum of 2 integers: 2+1, 1+2)
P(3, 3) = P(2, 2) + P(1, 2) + P(0, 2) = 1 + 0 + 0 = 1 (1 way to write 3 as an ordered sum of 3 integers: 1+1+1)
The total number of ways to write 3 as an ordered sum of positive integers is the sum of these counts: 1 + 2 + 1 = 4.
Similarly, you can use this approach to calculate the number of ways to write any given value of "n" as an ordered sum of positive integers.