SOLUTION: 3. Happy Harold’s Autos collected data on cars sold in October. 43 Total number of cars sold 23 Red cars 25 Cars with Air Conditioning 29 Cars with Heated Seats 17 Red car

Question 1195171: 3.
Happy Harold’s Autos collected data on cars sold in October.
43 Total number of cars sold
23 Red cars
25 Cars with Air Conditioning
29 Cars with Heated Seats
17 Red cars with Air Conditioning
15 Red cars with Heated Seats
16 Cars with Air Conditioning and Heated Seats
10 Red cars with Air Conditioning and Heated Seats
(a) Draw a Venn diagram illustrating all of the above information.
(b) How many red cars had neither Air Conditioning nor Heated Seats?
(c) How many cars had Airconditioning were not red?
(d) How many cars were not red and had neither Air Conditioning nor Heated Seats?

Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
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You are given the universal set U of 43 cars and 3 its basic subsets R, A and H (see the table below). U 43 total cars R 23 Red cars A 25 cars with Air conditioning H 29 cars with Heated seats Also, you are given info about their in-pair intersections and about their triple intersection. RA 17 Red cars with Air conditioning RH 15 Red cars with Heated seats AH 16 cars with Air conditioning and Heated seats RAH 10 Red cars with Air conditioning and Heated seats Having this info well organized, you can easily answer all questions (b), (c) and (d) without drawing Venn diagram. (b) The set {Red cars had neither Air Conditioning nor Heated Seats} is R \ ((RA U RH) \ RAH) and their number is 23 - (17 + 15 - 10) = 1. ANSWER (c) The set {cars had Air Conditioning were not red} is A \ RA, and their number is 25 - 17 = 8. ANSWER (d) The set {cars were not red and had neither Air Conditioning nor Heated Seats} is U \ (R U A U H). So, calculate the number of cars in the union (R U A U H} first. For it, use the inclusion-exclusion princuple/(formula) n(R U A U H) = n(R) + n(A) + n(H) - n(RA) - n(RH) - n(AH) + n(RAH) = = 23 + 25 + 29 - 17 - 15 - 16 + 10 = 39. Now the last step gives the answer to question (d) : the number of cars were not red and had neither Air Conditioning nor Heated Seats = 43 - 39 = 4. ANSWER

Solved.

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On inclusion-exclusion principle, see this Wikipedia article

https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle

To see many other similar (and different) solved problems, see the lessons

    - Counting elements in sub-sets of a given finite set
    - Advanced problems on counting elements in sub-sets of a given finite set
    - Challenging problems on counting elements in subsets of a given finite set
    - Selected problems on counting elements in subsets of a given finite set
    - Inclusion-Exclusion principle problems

in this site.

Happy learning (!)

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

3.
Happy Harold’s Autos collected data on cars sold in October.
43 Total number of cars sold
23 Red cars
25 Cars with Air Conditioning
29 Cars with Heated Seats
17 Red cars with Air Conditioning
15 Red cars with Heated Seats
16 Cars with Air Conditioning and Heated Seats
10 Red cars with Air Conditioning and Heated Seats
(a) Draw a Venn diagram illustrating all of the above information.
(b) How many red cars had neither Air Conditioning nor Heated Seats?
(c) How many cars had Airconditioning were not red?
(d) How many cars were not red and had neither Air Conditioning nor Heated Seats?

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