Problem
The members of a board of directors have the following roles: president $(P)$, vice president $(V)$, secretary $(S)$, treasurer $(T)$, and fundraiser $(F)$, Consider these board members to be a population of interest. The possible samples (without replacement) of size two that can be obtained from these five board members are as follows. P,V P,S P,T P,F V,S V,T V,F S,T S,F T,F If a simple random sampling method is used to obtain a sample of two of the board members, what are the chances of selecting the secretary and the treasurer? A. $\frac{2}{5}$ B. $\frac{1}{5}$ c. $\frac{1}{10}$ D. $\frac{1}{20}$
Solution
Step 1 ：Calculate the total number of ways to select 2 members out of 5 without replacement: \( \binom{5}{2} = 10 \)
Step 2 ：Calculate the number of ways to select the secretary and treasurer specifically: \( 1 \)
Step 3 ：Calculate the probability of selecting the secretary and treasurer: \( \frac{1}{10} \)
Step 4 ：Final Answer: \( \boxed{\frac{1}{10}} \)
