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Chapter 9: Problem 1

Find all partitions of \(7 .\)

Short Answer

Expert verified
The partitions of 7 are: \[ 7, 6 + 1, 5 + 2, 4 + 3, 5 + 1 + 1, 4 + 2 + 1, 3 + 3 + 1, 3 + 2 + 2, 4 + 1 + 1 + 1, 3 + 2 + 1 + 1, 2 + 2 + 2 + 1, 3 + 1 + 1 + 1 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1 \].

Step by step solution

01

Start Partitioning

Start with 7 itself. This is considered as the number 7 partitioned into one part. So, one partitioning is \[7\]. Then start breaking down the number into smaller parts.
02

Continue Breaking Down

One way to partition 7 is into two parts: \[ 7 = 6 + 1 = 5 + 2 = 4 + 3 \]. In this step, care should be taken to ensure that all combinations with two integers are grouped differently and not repeated.
03

Partition Into Three Parts

Partitioning 7 into three parts gives the combinations: \[ 7 = 5 + 1 + 1 = 4 + 2 + 1 = 3 + 3 + 1 = 3 + 2 + 2 \]. This results in 1 + 1 + 5, 1 + 2 + 4, 1 + 3 + 3, and 2 + 2 + 3 distinct sets.
04

Proceed with More Parts

Following the same approach, partitioning 7 into four, five, six, and seven parts can be worked out. One will find that when partitioned into seven parts, the only possibility is \[ 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 \].
05

Arrange All Partitions

Arrange all the distinct ways of partitioning 7 that were obtained in steps 1 through 4. Now, all partitions of 7 have been found.

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Most popular questions from this chapter

Carol is collecting money from her cousins to have a party for her aunt. If eight of the cousins promise to give \(\$ 2, \$ 3, \$ 4\), or \(\$ 5\) each, and two others each give \(\$ 5\) or \(\$ 10\), what is the probability that Carol will collect exactly \(\$ 40 ?\)

Find the coefficient of \(x^{20}\) in \(\left(x^{2}+x^{3}+x^{4}+x^{5}+x^{6}\right)^{5}\).

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a) Find the coefficient of \(x^{7}\) in $$ \left(1+x+x^{2}+x^{3}+\cdots\right)^{15} $$ b) Find the coefficient of \(x^{7}\) in \(\left(1+x+x^{2}+x^{3}+\cdots\right)^{n}\) for \(n \in \mathbf{Z}^{+}\)
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