How many customers purchase life or health but not auto insurance? An insurance agent sells life, health and auto insurance. During the year she met with 85 potential clients. Of these, 42 purchased life insurance, 40 health insurance, 24 auto insurance, 14 both life and health, 9 both life and auto, 11 both health and auto, and 2 purchased all three. How many of these potential clients purchased (a). no policies; (b) only health policies; (c) exactly one type of insurance; (d) life or health but not auto insurance?
A).
N(L or H or A) = N(L) + N(H) + N(A) – N(L and H) – N(L and A) – N(H and A) + N(L and H and A)
N(L or H or A) =  42+40+24-14-9-11 +2 -> 74
1 – N(L or H or A) = Number who did not get any policies
85 – 74 -> 11 = number without policies
B). N(H) = 40, so subtract overlaps and add back overlap 40 – 14 – 11  + 2 = 17
c). [N(L) – N(L and H) – N(L and A) + N(L and H and A)] + [N(H) – N(L and H) – N(H and A) + N(L and H and A)] + [N(A) – N(H and A) – N(L and A) + N(H and A and L)]
[40-14-11+2] +[42-14-9+2] + [24-9-11+2] = 44
d). (40-14) + (42 - 14) = 54 (my answer) but the book says the answer is 50.
I've done and understand how A,b,c are done but not quite sure about the intuition regarding (d). Would appreciate if someone can point me in the right direction!
 A: Venn Diagrams are your friend!


*

*Draw the Venn Diagram for the 3 sets:





*Start in the middle (2 purchase all 3 types of insurance)





*Find other intersections (9 Auto and Life; 11 Auto and Health; 14 Life and Health)





*Complete Insides (42 Life; 40 Health; 24 Auto):





*And the number on the outside (85 clients total):



With this one diagram you can now answer all questions in a flash:
a) No insurance: 11. Proof: See diagram!
b) Only Health: 17. Proof: See diagram!
c) Exactly 1 type: 6+21+17=44 Proof: See diagram!
d) Life or Health but not Auto: 21+12+17=50. Proof: See diagram!
A: Let $|L|$, $|H|$, and $|A|$ denote, respectively, the number of customers who purchased life insurance, health insurance, and auto insurance.  The number of customers who purchase health insurance or life insurance is 
$$|H \cup L| = |H| + |L| - |H \cap L| = 40 + 42 - 14 = 68$$
The number of life insurance or health insurance customers who have also purchased auto insurance is 
$$|A \cap H| + |A \cap L| - |A \cap H \cap L| = 9 + 11 - 2 = 18$$
Hence, the number of customers who purchase health insurance or life insurance but not auto insurance is 
$$|H \cup L| - (|A \cap H| + |A \cap L| - |A \cap H \cap L|) = 68 - 18 = 50$$   
