Question

A partition of a positive integer $n$ is any way of writing $n$ as a sum of one or more positive integers, in which we don't care about the order of the numbers in the sum. for example, the number 4 can be written as a sum of one or more positive integers (where we don't care about the order of the numbers in the sum) in exactly five ways: \[4,\; 3 + 1,\; 2 + 2,\; 2 + 1 + 1,\; 1 + 1 + 1 + 1.\] so 4 has five partitions. what is the number of partitions of the number 7? aops

Answer 1

To find the number of partitions of the number 7, we list all unique combinations of integers that sum to 7. Counting these combinations, we determine that there are 15 partitions of the number 7.

Step-by-step explanation:

The partition of a positive integer n is the number of ways you can write n as a sum of one or more positive integers, disregarding the order of the addends. To find the partitions of 7, we list out all possible combinations:

• 7
• 6 + 1
• 5 + 2
• 5 + 1 + 1
• 4 + 3
• 4 + 2 + 1
• 4 + 1 + 1 + 1
• 3 + 3 + 1
• 3 + 2 + 2
• 3 + 2 + 1 + 1
• 3 + 1 + 1 + 1 + 1
• 2 + 2 + 2 + 1
• 2 + 2 + 1 + 1 + 1
• 2 + 1 + 1 + 1 + 1 + 1
• 1 + 1 + 1 + 1 + 1 + 1 + 1

Counting all the combinations, there are 15 partitions of the number 7.

Answer 2

1*7
1+1+1+1+1+1+1
[7]
5+2
10-3