I need help with this discrete math question

Question

Hi, i need a little help with this equation I'm stuck on.

If there are

65 students taking french,

45 taking German,

42 taking italian,

20 taking both french and german,

25 taking both french and italian,

15 taking both German and Italian, and

8 taking all 3 languages, how many students are there?

Answer 1

Make a diagram with three mutually intersecting circles. It's impossible to do here.

Look at the clues in reverse order

The intersection of the 3 circles are 8 students taking all 3.

15 are taking German and Italian, including the 8 who are also taking French. So 7 students are taking German and Italian, but not French.

25 are taking French and Italian, including the 8 who are taking all 3. That leaves 17 who are taking only French and Italian.

20 are taking French and German, including the 8 who are also taking Italian, leaving 12 who are taking only French and German.

42 are taking Italian, including 8 taking all 3, and 7 taking German as well, and 17 taking French and Italian. That leaves 10 who are only taking Italian.

45 are taking German, including 8 all 3, 12 F&G, and 7 G&I, leaving 18 taking German alone.

65 are taking French, including 8 all 3, 17 F&I, and 12 F&G, leaving 28 taking French alone.

The total # of students is 100, assuming I have no typos.

28 F only

12 F&G only

18 G only

7 G&I only

10 I only

17 F&I only

8 all 3 languages.