2. All the students in SS3 of a named school take either Mathematics (M), or Physics (P) or Chemistry (C). 40 take Mathematics, 42 take physics, 38 take Chemistry, 20 take Mathematics and Physics, 28 take Physics and Chemistry while 25 take mathematics and chemistry.
How many take
(a) All the three subject: 120
(b) Mathematics, but neither Physics nor Chemistry
(c) Physics, but neither Mathematics nor Chemist
To find the number of students who take all three subjects:
Step 1: Start with the total number of students in SS3:
Total students = Number of students taking Mathematics + Number of students taking Physics + Number of students taking Chemistry
Step 2: Use the given information to find individual counts:
Mathematics (M) = 40
Physics (P) = 42
Chemistry (C) = 38
Step 3: Use the given information about students taking multiple subjects:
Mathematics and Physics (M and P) = 20
Physics and Chemistry (P and C) = 28
Mathematics and Chemistry (M and C) = 25
Step 4: Create equations using the principle of inclusion-exclusion:
Total students = M + P + C - (M and P) - (P and C) - (M and C) + (M, P, and C)
Step 5: Plug in the given numbers into the equation:
Total students = 40 + 42 + 38 - 20 - 28 - 25 + (M, P, and C)
Total students = 107 + (M, P, and C)
Step 6: Solve for (M, P, and C) by rearranging the equation:
(M, P, and C) = Total students - 107
(M, P, and C) = 107 - 107
(M, P, and C) = 0
Therefore, no student takes all three subjects.
To find the number of students taking Mathematics but neither Physics nor Chemistry:
Step 1: Use the given information to find individual counts:
Mathematics (M) = 40
Mathematics and Physics (M and P) = 20
Mathematics and Chemistry (M and C) = 25
Step 2: Use the principle of inclusion-exclusion:
Mathematics Only = M - (M and P) - (M and C)
Mathematics Only = 40 - 20 - 25
Mathematics Only = -5
Since the count is negative, it means there might be an error or contradiction in the given information. In this case, it is not possible to determine the number of students taking Mathematics but neither Physics nor Chemistry.
To find the number of students taking Physics but neither Mathematics nor Chemistry:
Step 1: Use the given information to find individual counts:
Physics (P) = 42
Mathematics and Physics (M and P) = 20
Physics and Chemistry (P and C) = 28
Step 2: Use the principle of inclusion-exclusion:
Physics Only = P - (M and P) - (P and C)
Physics Only = 42 - 20 - 28
Physics Only = -6
Similar to the previous case, the count is negative, indicating a potential error or contradiction. Thus, it is not possible to determine the number of students taking Physics but neither Mathematics nor Chemistry.