# How many subsets of size three or less are in a n-object set

• Eclair_de_XII
In summary: So I guess the answer is the same for all three questions then. In summary, the answer is ##\binom n k## for ##k\leq 3##, where ##n## is the total number of applicants and ##k## is the size of the subset chosen for the job.
Eclair_de_XII

## Homework Statement

(a) How many ways can at most three people out of a selection of ##n## applicants be selected for a job?
(b) How many subsets of size at most three are there in a set of size ##n##?
(c) How many ways can a given subset of size three or fewer be chosen for the job?

## The Attempt at a Solution

(a) There are ##\sum_{i=0}^3 \binom n i## ways.
(b) Define the set of n objects as ##A##. Then define a subset ##\sigma_i=\{x\in P(A): |x|\leq 3\}##. Now define a set of integers ##J=\{i\in \mathbb{N}:\sigma_i\cap \sigma_j=∅,\sum_i |\sigma_i|=n\}##. Now define ##S_k=\cup_{j \in J} \sigma_j##, and so for a given selection of subsets, the number of subsets of at most size three is: ##\frac{n!}{\Pi_{\sigma_i\in S_k} |\sigma_i|!}##. And so the total number of possible subsets of at most size three is: ##\sum_{k=1}^{|\{\text{possible arrangements of subsets of at most three}\}|}\frac{n!}{\Pi_{\sigma_i\in S_k} |\sigma_i|!}##

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Do you think (b) or (c) differs from (a) other than in how they are expressed? If so, how? (Note that in none of the three cases are the subsets specified to be ordered).

andrewkirk said:
Do you think (b) or (c) differs from (a) other than in how they are expressed? If so, how?

Well, in part (a), you're choosing one subset of at most size three to be selected for a job, while part (b) asks for how many of these possible subsets there are in the n-size set. If I had to hazard a guess, I'd say that they're both the same, in the respect that ##\binom n k## for ##k\leq 3## counts how many subsets of ##k## elements can be formed from a set of ##n## objects. So I'll take a shot in the dark and say that the answer to part (a) is the same as in part (b), or part (b)'s answer looks something like: ##\binom n k## for ##k\leq 3##.

Eclair_de_XII said:
Well, in part (a), you're choosing one subset of at most size three to be selected for a job, while part (b) asks for how many of these possible subsets there are in the n-size set. If I had to hazard a guess, I'd say that they're both the same, in the respect that ##\binom n k## for ##k\leq 3## counts how many subsets of ##k## elements can be formed from a set of ##n## objects. So I'll take a shot in the dark and say that the answer to part (a) is the same as in part (b), or part (b)'s answer looks something like: ##\binom n k## for ##k\leq 3##.
Yes, the set of applicants is a set of size ##n## and the set of people selected for the job is a subset of that, of size at most three.

The three questions are the same, just expressed differently.

Huh, thanks.

## 1. How do you calculate the number of subsets of size three or less in a set with n objects?

The number of subsets of size three or less in a set with n objects can be calculated using the formula 2^n - 1. This formula takes into account the empty set and all possible combinations of the n objects, including subsets of size three or less.

## 2. Can you give an example of a set with n objects and its corresponding number of subsets of size three or less?

For a set with n = 4 objects, the number of subsets of size three or less would be 2^4 - 1 = 15. These subsets could include {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, and the empty set.

## 3. How does the number of subsets of size three or less change as n increases?

As n increases, the number of subsets of size three or less also increases exponentially. For example, a set with n = 5 objects would have 2^5 - 1 = 31 subsets of size three or less.

## 4. Is there a limit to the number of subsets of size three or less in a set with n objects?

Yes, there is a limit. The maximum number of subsets of size three or less in a set with n objects is 2^n - 1. This limit is reached when all possible combinations of the n objects have been included in the subsets.

## 5. How is the concept of subsets of size three or less useful in real-world applications?

The concept of subsets of size three or less is useful in various fields of science, such as genetics, computer science, and statistics. It allows for the analysis and categorization of data, as well as the identification of patterns and relationships within a set of objects. In genetics, for example, subsets of genes can be studied to understand their role in a particular trait or disease. In computer science, subsets of data can be used for data mining and machine learning algorithms. And in statistics, subsets can be used to create representative samples for research studies.

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