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:
(b) Mathematics, but neither Physics nor Chemistry
(c) Physics, but neither Mathematics nor Chemist
(a) All the three subjects:
To find the number of students who take all three subjects, we need to subtract the number of students who take exactly two subjects from the total number of students who take Mathematics, Physics, and Chemistry.
Let's use the principle of inclusion-exclusion to find the answer:
Total students who take Mathematics (M) = 40
Total students who take Physics (P) = 42
Total students who take Chemistry (C) = 38
Total students who take Mathematics and Physics (M&P) = 20
Total students who take Physics and Chemistry (P&C) = 28
Total students who take Mathematics and Chemistry (M&C) = 25
Total number of students who take at least two subjects: M&P + P&C + M&C - (2 * All three) = 20 + 28 + 25 - (2 * All three)
All three = (20 + 28 + 25 - (2 * All three))
Simplifying the equation, we get:
All three = 20 + 28 + 25 - 2(All three)
3(All three) = 73
All three = 73/3
All three = 24.33 (rounding down to 24)
Therefore, the number of students who take all three subjects is 24.
(b) Mathematics, but neither Physics nor Chemistry:
To find the number of students who take Mathematics but neither Physics nor Chemistry, we need to subtract the number of students who take Mathematics and Physics, Mathematics and Chemistry, and Mathematics, Physics, and Chemistry from the total number of students who take Mathematics.
Total students who take Mathematics (M) = 40
Total students who take Mathematics and Physics (M&P) = 20
Total students who take Mathematics and Chemistry (M&C) = 25
Total students who take all three subjects (All three) = 24
Mathematics, but neither Physics nor Chemistry = M - (M&P + M&C + All three)
Mathematics, but neither Physics nor Chemistry = 40 - (20 + 25 + 24)
Mathematics, but neither Physics nor Chemistry = 40 - 69
Mathematics, but neither Physics nor Chemistry = -29
Since the result is negative, there are no students who take Mathematics but neither Physics nor Chemistry.
(c) Physics, but neither Mathematics nor Chemistry:
To find the number of students who take Physics but neither Mathematics nor Chemistry, we need to subtract the number of students who take Physics and Mathematics, Physics and Chemistry, and Mathematics, Physics, and Chemistry from the total number of students who take Physics.
Total students who take Physics (P) = 42
Total students who take Mathematics and Physics (M&P) = 20
Total students who take Physics and Chemistry (P&C) = 28
Total students who take all three subjects (All three) = 24
Physics, but neither Mathematics nor Chemistry = P - (M&P + P&C + All three)
Physics, but neither Mathematics nor Chemistry = 42 - (20 + 28 + 24)
Physics, but neither Mathematics nor Chemistry = 42 - 72
Physics, but neither Mathematics nor Chemistry = -30
Since the result is negative, there are no students who take Physics but neither Mathematics nor Chemistry.
To find the number of students who take all three subjects, we can use the principle of inclusion-exclusion.
Given:
Total number of students taking Mathematics (M) = 40
Total number of students taking Physics (P) = 42
Total number of students taking Chemistry (C) = 38
Number of students taking Mathematics and Physics (M&P) = 20
Number of students taking Physics and Chemistry (P&C) = 28
Number of students taking Mathematics and Chemistry (M&C) = 25
(a) To find the number of students taking all three subjects (M, P, and C), we can subtract the number of students who only take Mathematics, Physics, or Chemistry from the total number of students in SS3.
Total number of students in SS3 = 40 + 42 + 38 = 120
Number of students taking all three subjects = Total number of students in SS3 - (Number of students taking only Mathematics + Number of students taking only Physics + Number of students taking only Chemistry + Number of students taking exactly two subjects)
Number of students taking all three subjects = 120 - (40 + 42 + 38 - 20 - 28 - 25)
= 120 - (115 - 73)
= 120 - 42
= 78
Therefore, 78 students take all three subjects (Mathematics, Physics, and Chemistry).
(b) To find the number of students taking Mathematics but neither Physics nor Chemistry, we need to subtract the students taking both Mathematics and Physics, and the students taking both Mathematics and Chemistry from the total number of students taking Mathematics.
Number of students taking Mathematics but neither Physics nor Chemistry = Number of students taking Mathematics - Number of students taking both Mathematics and Physics - Number of students taking both Mathematics and Chemistry
Number of students taking Mathematics but neither Physics nor Chemistry = 40 - 20 - 25
= 40 - 45
= -5
Since the resulting value is negative, it means that there are no students taking Mathematics but neither Physics nor Chemistry.
(c) To find the number of students taking Physics but neither Mathematics nor Chemistry, we need to subtract the students taking both Physics and Chemistry, and the students taking both Physics and Mathematics from the total number of students taking Physics.
Number of students taking Physics but neither Mathematics nor Chemistry = Number of students taking Physics - Number of students taking both Physics and Chemistry - Number of students taking both Physics and Mathematics
Number of students taking Physics but neither Mathematics nor Chemistry = 42 - 28 - 20
= 42 - 48
= -6
Since the resulting value is negative, it means that there are no students taking Physics but neither Mathematics nor Chemistry.
Therefore, there are 78 students taking all three subjects (Mathematics, Physics, and Chemistry), and there are no students taking Mathematics but neither Physics nor Chemistry, and no students taking Physics but neither Mathematics nor Chemistry.
To find the number of students who take all three subjects (a), we need to find the intersection between the sets of students taking Mathematics, Physics, and Chemistry.
The formula we will use is:
n(A ∩ B ∩ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)
Now let's substitute the values given in the problem:
n(A) = 40 (students taking Mathematics)
n(B) = 42 (students taking Physics)
n(C) = 38 (students taking Chemistry)
n(A ∩ B) = 20 (students taking both Mathematics and Physics)
n(A ∩ C) = 25 (students taking both Mathematics and Chemistry)
n(B ∩ C) = 28 (students taking both Physics and Chemistry)
Plugging in the values, we get:
n(A ∩ B ∩ C) = 40 + 42 + 38 - 20 - 25 - 28 + n(A ∩ B ∩ C)
Simplifying the equation gives us:
n(A ∩ B ∩ C) = 47
Therefore, 47 students take all three subjects (Mathematics, Physics, and Chemistry).
To find the number of students taking Mathematics, but neither Physics nor Chemistry (b), we need to find the students taking Mathematics but not taking Physics and Chemistry, which is the difference between the students taking Mathematics and the intersection of Mathematics with Physics and Chemistry.
Using the formula:
n(A - B - C) = n(A) - n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩ C)
Plugging in the values:
n(A - B - C) = 40 - 20 - 25 + 47
Simplifying the equation gives us:
n(A - B - C) = 42
Therefore, 42 students take Mathematics but neither Physics nor Chemistry.
To find the number of students taking Physics, but neither Mathematics nor Chemistry (c), we need to find the students taking Physics but not taking Mathematics and Chemistry, which is the difference between the students taking Physics and the intersection of Physics with Mathematics and Chemistry.
Using the same formula as before:
n(B - A - C) = n(B) - n(A ∩ B) - n(B ∩ C) + n(A ∩ B ∩ C)
Plugging in the values:
n(B - A - C) = 42 - 20 - 28 + 47
Simplifying the equation gives us:
n(B - A - C) = 41
Therefore, 41 students take Physics but neither Mathematics nor Chemistry.