My cat keeps to himself most of the time. I only heard him meow, hiss, and purr on one day
out of the last 23 days. But I did hear him make at least one of these sounds each day. I heard
him hiss on 6 days, purr on 12 days, and meow on 15 days. On 2 days, I heard him meow and
hiss but not purr, and on 2 days, I heard him purr and hiss but not meow. On how many days
did I hear him meow and purr but not hiss?  Pick: A) 1 B) 2 C) 3 D) 4 E) 5
Let M denote the days on which the cat meowed, H the days on which he hissed,   and P the days on which he purred. Given the data, the following Venn diagram represents   the situation, where the expressions inside the circles are the cardinalities of disjoint sections:   We are looking for x. The labeled sections of the Venn diagram are disjoint sets, so they   should add to 23. That is,   23 = 1 + 1 + 2 + 2 + x + (9 􀀀 x) + (12 􀀀 x) = 27 􀀀 x:   Therefore,x = 4.   or   The inclusion-exclusion principle gives   23 = jM [ H [ Pj = jMj + jHj + jPj 􀀀 jM \ Hj 􀀀 jH \ Pj 􀀀 jM \ Pj + jM \ H \ Pj   = 15 + 6 + 12 􀀀 (2 + 1) 􀀀 (2 + 1) 􀀀 (x + 1) + 1:   So x = 4.   correct answer 4
<result>D</result>
My cat keeps to himself most of the time. I only heard him meow, hiss, and purr on one day
out of the last 23 days. But I did hear him make at least one of these sounds each day. I heard
him hiss on 6 days, purr on 12 days, and meow on 15 days. On 2 days, I heard him meow and
hiss but not purr, and on 2 days, I heard him purr and hiss but not meow. On how many days
did I hear him meow and purr but not hiss?