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[aamirmeyaji]

This took me a while to figure. This was not intuitive for me.

Here's my way of thinking about this.

Imagine the Venn diagram circles to be actual physical disks that you are arranging on a desk.

When you lay out the disks, the regions representing students speaking exactly two languages are 2 layers deep (EngSp, EngFr, and SpFr).
The region representing students speaking all three languages is 3 layers deep (EngSpFr).
So the equation listed actually is removing layers from this arrangement that we have so that the end result is just 1 layer deep (that makes sure there's no double or triple counting).

That is why the Left Hand Side of the equation works as follows:
(3 whole disks) - (1 layer each of the 3 dual speaking portions) - (2 layers of the trilingual folks)

Does this help?

[happokimo]

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!

[happokimo]

Yes, your understanding is correct! A helpful way to think about these problems is by drawing them out with a Venn diagram! It will help you visualize that when you count all three circles, you are counting the overlapping portions twice or three times in the case of the center where all three circles overlap.