Question

To join a certain club, a person must be either a statistician or a mathematician or both. Of the 25 members in this club, 19 are statisticians and 16 are mathematicians. How many persons in the club are both a statistician and a mathematician?

Short answer

The number of people who are both statisticians and mathematicians is 10.

Step-by-step solution

Identify the Variables

In this problem, there are several variables to identify: - The total number of members of the club: 25 - The number of statisticians in the club: 19 - The number of mathematicians in the club: 16 - The unknown number of individuals who are both statisticians and mathematicians. This last number is what we're trying to solve for.

Apply the Formula for the Union of Two Sets

The formula for the union of two sets is \( n(A ∪ B) = n(A) + n(B) - n(A ∩ B) \). Plugging in the given variables gives: 25 (which is \( n(A ∪ B) \) ) = 19 (which is \( n(A) \) ) + 16 (which is \( n(B) \) ) - \( n(A ∩ B) \).

Rearrange the Formula to Solve for n(A ∩ B)

To isolate \( n(A ∩ B) \) on one side of the equation, subtract the number of each individual set from both sides. The equation after doing that is: \( n(A ∩ B) = n(A) + n(B) - n(A ∪ B) = 19 + 16 - 25 \).

Calculate the Value of n(A ∩ B)

When the numbers are added and subtracted on the right side of the equation, \( n(A ∩ B) \) equals 10. This means that there are 10 members of the club who are both statisticians and mathematicians.