Exercise · ZW-INTEGER-PARTITION-006 · ★★☆ Intermediate
How many partitions does \(5\) have?
Choose one option.
Show answer
- B. \(7\)
Explanation
The partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1 and 1+1+1+1+1, that is seven. The option giving six omits one of these forms.
Review the concept: Integer partition
A partition of an integer n is a way of writing n as a sum of positive integers, where the order of the summands does not matter. The number of partitions of n is denoted p(n).
\[\sum_{n\ge 0} p(n)x^n=\prod_{k\ge 1}\frac{1}{1-x^k}\]
More practice problems: integer partition
- ZW-INTEGER-PARTITION-001 How many partitions does the integer 3 have?
- ZW-INTEGER-PARTITION-002 Which of the following is a partition of 5?
- ZW-INTEGER-PARTITION-003 How many partitions of 4 have exactly two parts?
- ZW-INTEGER-PARTITION-004 What is the number of partitions p(2)?
- ZW-INTEGER-PARTITION-005 Which partition of 6, written in non-increasing order, has all its parts equal to 2?
- ZW-INTEGER-PARTITION-007 How many partitions of 6 have all parts distinct?
- ZW-INTEGER-PARTITION-008 What is the coefficient of x^3 in prod_k≥ 1(1)/(1-x^k)?
- ZW-INTEGER-PARTITION-009 How many partitions of 4 have all parts odd?