Question
10 (2 marks) A Thai restaurant offers a menu containing 65 dishes, which includes dishes with
chicken, pork or prawn. The number of chicken dishes is 15 , the number of pork dishes
is 15 and the number of prawn dishes is 12 . If there are 25 dishes that do not contain any
of the three ingredients, determine the number of dishes on the menu that contain chicken
or pork, but not prawn.
chicken, pork or prawn. The number of chicken dishes is 15 , the number of pork dishes
is 15 and the number of prawn dishes is 12 . If there are 25 dishes that do not contain any
of the three ingredients, determine the number of dishes on the menu that contain chicken
or pork, but not prawn.
Ask by Savage Chan. in Australia
Feb 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
28 dishes contain chicken or pork but not prawn.
Solution
We are given that the restaurant’s menu has 65 dishes in all. Out of these, 25 have none of the three ingredients (chicken, pork, or prawn) so that
Dishes with at least one ingredient = 65 – 25 = 40.
Let the sets C, P, and R be the dishes containing chicken, pork, and prawn respectively. We are told
|C| = 15, |P| = 15, |R| = 12.
The formula for the union of three sets is
|C ∪ P ∪ R| = |C| + |P| + |R| − (|C ∩ P| + |C ∩ R| + |P ∩ R|) + |C ∩ P ∩ R|.
Plug in what we know:
40 = 15 + 15 + 12 − (|C ∩ P| + |C ∩ R| + |P ∩ R|) + |C ∩ P ∩ R|.
Simplify the numbers:
40 = 42 − (|C ∩ P| + |C ∩ R| + |P ∩ R|) + |C ∩ P ∩ R|.
Rearrange to obtain a useful relation among the intersections:
|C ∩ P| + |C ∩ R| + |P ∩ R| − |C ∩ P ∩ R| = 42 − 40 = 2. (1)
We wish to count the dishes that contain chicken or pork but not prawn. In set notation, we want
|(C ∪ P) \ R|.
Notice that
|(C ∪ P) \ R| = |C ∪ P| − |(C ∪ P) ∩ R|.
We can compute |C ∪ P| by
|C ∪ P| = |C| + |P| − |C ∩ P| = 15 + 15 − |C ∩ P| = 30 − |C ∩ P|.
Next, note that a dish that is both (chicken or pork) and prawn is one that belongs to (C ∩ R) or (P ∩ R). Thus
|(C ∪ P) ∩ R| = |C ∩ R| + |P ∩ R| − |C ∩ P ∩ R|.
Thus the count we want is
|(C ∪ P) \ R| = (30 − |C ∩ P|) − [|C ∩ R| + |P ∩ R| − |C ∩ P ∩ R|].
Distribute the negative sign:
|(C ∪ P) \ R| = 30 − |C ∩ P| − |C ∩ R| − |P ∩ R| + |C ∩ P ∩ R|.
Now look at the combination of intersections in (1). We found
|C ∩ P| + |C ∩ R| + |P ∩ R| − |C ∩ P ∩ R| = 2.
It follows that
(|C ∩ P| + |C ∩ R| + |P ∩ R|) = 2 + |C ∩ P ∩ R|.
Substitute this into the expression for |(C ∪ P) \ R|:
|(C ∪ P) \ R| = 30 − [2 + |C ∩ P ∩ R|] + |C ∩ P ∩ R|.
Notice that –|C ∩ P ∩ R| and +|C ∩ P ∩ R| cancel out, leaving
|(C ∪ P) \ R| = 30 − 2 = 28.
Thus, 28 dishes on the menu contain chicken or pork but not prawn.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Mind Expander
To solve the problem, we first note the total number of dishes on the menu (65) and the number that do not contain chicken, pork, or prawn (25). This means that the number of dishes containing one or more of chicken, pork, or prawn is:
65 - 25 = 40 dishes.
We have:
- Chicken dishes = 15
- Pork dishes = 15
- Prawn dishes = 12
Let’s denote the number of dishes that contain both chicken and pork as
The principle of inclusion-exclusion gives us:
Substituting in the values:
This gives us:
Rearranging yields:
Next, we need to find the number of dishes that contain chicken or pork but not prawn. These dishes can be found with the equation:
However, further information on the overlaps is necessary, which isn’t provided. Given that:
we can simplify further. But again we hit a wall without the additional overlaps
Hence, instead of heading further into unknowns, let’s make a key realization:
The total number of dishes with chicken or pork (excluding prawn) must also adhere to:
From this understanding, without specified overlaps, a reasonable feasible outcome given the stated constraints could be determined through logical deduction around what the minimal necessary overlaps will yield. Specifically,
Assuming x = 0 yields:
Thus, in summary, the dishes containing chicken or pork, but not prawn, is flexible based on overlaps but will be at least greater than or equal to:
30 - Adjustments for overlaps (if any).
With a little assumption play through using basic deductions of combinatorial placement without firm elements placed in specific ratios due to unavailability of full shared inclusions.
In this case, if we assume minimal overlap scenario values yielding the least constraint and maximum actual piece values being provided, we can thus feasibly provide that:
Answer: 30 - overlap values.
