Question

at a college, 56 students were members of clubs or the student council, 39 were members of clubs, and 10 were members of clubs and the student council. how many students were members only of the student council? (type a whole number.)

Explanation:

Step1: Use the principle of inclusion - exclusion

Let \(A\) be the set of club - members and \(B\) be the set of student - council members. We know that \(n(A\cup B)=56\), \(n(A) = 39\), and \(n(A\cap B)=10\).
We want to find \(n(B\setminus A)\) (students who are only in the student council).
By the formula \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\), we can first find \(n(B)\) as \(n(B)=n(A\cup B)-n(A)+n(A\cap B)\).
Then the number of students who are only in the student council is \(n(B)-n(A\cap B)\).

Step2: Calculate \(n(B)\)

Substitute the given values into the formula \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\).
\[n(B)=n(A\cup B)-n(A)+n(A\cap B)=56 - 39+10=27\]

Step3: Calculate the number of students only in the student council

The number of students who are only in the student council is \(n(B)-n(A\cap B)\).
\[n(B)-n(A\cap B)=27 - 10=17\]

Answer:

17