Inclusion and Exclusion Mouthwash Problem I am working on a problem for my Discrete Math class. Nineteen different mouthwash products make the following claims: 12 claim to freshen breath, 10 claim to prevent gingivitis, 11 claim to reduce plaque, 6 claim to both freshen breath and reduce plaque, 5 claim to both prevent gingivitis and freshen breath, and 5 claim to both prevent gingivitis and reduce plaque. a. How many products make all three claims? b. How many products claim to freshen breath but do not claim to reduce plaque? I got part A figured out as being 12+10+11-6-5-5=17, so the answer is 2. How do I find part B?

There are 12 which freshen breath, and there are 6 which freshen breath and reduce plaque too. Thus there are 12-6=6 which excludes plaque but includes only breath freshening paste.