Question

Of the 200 seniors at Monroe High School, exactly 40 are in the band, 60 are in the orchestra, and 10 are in both. How many seniors are in neither the band nor the orchestra?

Answer 1

Using the principle of inclusion-exclusion, we find that there are 110 seniors at Monroe High School who are in neither the band nor the orchestra. We calculate this by subtracting the number of students in either the band or orchestra (90) from the total number of seniors (200).

Step-by-step explanation:

To determine how many seniors at Monroe High School are in neither the band nor the orchestra, we can use the principle of inclusion-exclusion. The principle of inclusion-exclusion states that if we want to find the total number of items in two sets (in this case, the band and orchestra), we add the number of items in each set and then subtract the number of items that are in both sets.

We are given that there are 40 seniors in the band, 60 seniors in the orchestra, and 10 seniors in both the band and the orchestra. Let's denote the number of seniors in neither as 'N'.

First, we calculate the number of seniors who are in either the band or orchestra, or both:

• Number in band or orchestra = Number in band + Number in orchestra - Number in both
• Number in band or orchestra = 40 + 60 - 10
• Number in band or orchestra = 90

To find the number of seniors in neither, we subtract the number of seniors in the band or orchestra from the total number of seniors:

• N = Total number of seniors - Number in band or orchestra
• N = 200 - 90
• N = 110

So, there are 110 seniors at Monroe High School who are in neither the band nor the orchestra.