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**1. The problem statement, all variables and given/known data**

(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?

**2. Relevant equations**

**3. 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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