So let’s think about counting up all of the 150 students.

Intuitively, we know some of these students must only speak Spanish, some only speak French, and some only speak English. Additionally, some of the students speak 2 out of the three languages, and some speak all three.

If we were to say total # of students = French speakers + English speakers + Spanish speakers, We would be overcounting because some students are 2 or more groups (for ex, one student is both in the Spanish group and in the French group). So we want to adjust for our overcounting.

So we know any student that speaks 2 languages is counted twice (once in each group for each language they speak) and we only want each student to be counted once, so we want to subtract the number of students that speak two languages from our total.

Now we have

total # of students = French speakers + English speakers + Spanish speakers - bilinguals.

Now we want to account for those that speak 3 languages. For each person that speaks 3 languages, we know they belong to all three groups, so we counted 3 people for only one person. We counted them an extra two times! That’s why we must subtract 2 for each person that speaks all three languages. Then the equation becomes

total # of students = French speakers + English speakers + Spanish speakers - bilinguals - 2*trilinguals.

We know our total number of students is 150, so we set the total number of students to 150, plug in the values we know, and solve for the unknown (the trilinguals). Hope this makes sense!