# Set theory and Venn diagram

In a group of 70 cars tested by a garage in Delhi, 15 had faulty tyres, 20 had faulty brakes and 18 exceed the allowable emission limits. Also, 5 cars had faulty tyres and breaks, 6 failed on tyres and emission limits, 10 failed on breaks and emission, and 4 cars were unsatisfactory in all aspects, how many had no fault in these 3 checks?

• Hi Buchi, welcome to MSE! What have you tried? This is not a forum where people just do your HW for you. Please see math.meta.stackexchange.com/questions/9959/… The Venn diagram approach you allude to in the title is a good one! Commented Apr 17, 2019 at 20:28

Easy, we have 70 total subtract 15+20+18=53, giving 17. Then add back the overlaps of any 2 as they got subtracted twice, 6+10+5+17=38. Now deal with the threeway overlap, it got subtracted 3 times (not the 6 I thought, because the overlaps overlap), then added back 3 times. We need to subtract it once more. 38-4=34. 34 cars had no fault when checked,70-34=36 did. Next time show what's been tried, and the problem source.

• Are you taking 4 as the 3 overlaps? I thought they said 4 were unsatisfactory in all 3 aspects? Isn’t it meant to be excluded from all the three aspects? Commented Apr 18, 2019 at 10:11
• no, it means standards were not satified in all three areas.
– user645636
Commented Apr 18, 2019 at 10:12
• okay found a slight error in my math, but you can use this method for arbitrarily many sets.
– user645636
Commented Apr 18, 2019 at 10:17
• Is there a way I can snap my Venn diagram and the question and send you here? Commented Apr 18, 2019 at 10:19