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A class has 80 students. 45 take Physics, 40 take Chemistry, 35 take Biology. 18 take Physics and Chemistry, 15 take Physics and Biology, 12 take Chemistry and Biology, and 6 take all three. How many

Mathematics

ScanSolve AI Answer

Answer

81\text{81}

16\text{16}

27\text{27}

Step-by-step solution

9 steps

:

Let PP be the set of students who take Physics, CC for Chemistry, and BB for Biology.
Given:
Total students = 80
∣P∣=45|P| = 45
∣C∣=40|C| = 40
∣B∣=35|B| = 35
∣P∩C∣=18|P \cap C| = 18
∣P∩B∣=15|P \cap B| = 15
∣C∩B∣=12|C \cap B| = 12
∣P∩C∩B∣=6|P \cap C \cap B| = 6

1. Number taking at least one subject
This is equivalent to finding ∣P∪C∪B∣|P \cup C \cup B|. We use the Principle of Inclusion-Exclusion.

  1. Apply the Principle of Inclusion-Exclusion formula.

    ∣P∪C∪B∣=∣P∣+∣C∣+∣B∣−(∣P∩C∣+∣P∩B∣+∣C∩B∣)+∣P∩C∩B∣|P \cup C \cup B| = |P| + |C| + |B| - (|P \cap C| + |P \cap B| + |C \cap B|) + |P \cap C \cap B|

  2. Substitute the given values into the formula.

    ∣P∪C∪B∣=45+40+35−(18+15+12)+6|P \cup C \cup B| = 45 + 40 + 35 - (18 + 15 + 12) + 6

  3. Perform the calculations.

    ∣P∪C∪B∣=120−45+6|P \cup C \cup B| = 120 - 45 + 6 ∣P∪C∪B∣=75+6|P \cup C \cup B| = 75 + 6 ∣P∪C∪B∣=81|P \cup C \cup B| = 81

    The number of students taking at least one subject is 81\boxed{\text{81}}.

    2. Number taking only Chemistry
    This means students who take Chemistry but do not take Physics and do not take Biology.

  4. Use the formula for "only C".

    ∣C only∣=∣C∣−(∣P∩C∣+∣C∩B∣)+∣P∩C∩B∣|C \text{ only}| = |C| - (|P \cap C| + |C \cap B|) + |P \cap C \cap B|

  5. Substitute the given values.

    ∣C only∣=40−(18+12)+6|C \text{ only}| = 40 - (18 + 12) + 6

  6. Perform the calculations.

    ∣C only∣=40−30+6|C \text{ only}| = 40 - 30 + 6 ∣C only∣=10+6|C \text{ only}| = 10 + 6 ∣C only∣=16|C \text{ only}| = 16

    The number of students taking only Chemistry is 16\boxed{\text{16}}.

    3. Number taking exactly two subjects
    This is the sum of students taking (Physics and Chemistry only), (Physics and Biology only), and (Chemistry and Biology only).

  7. Calculate the number of students taking exactly two subjects for each pair.

    Number taking Physics and Chemistry only: ∣P∩C∣−∣P∩C∩B∣=18−6=12|P \cap C| - |P \cap C \cap B| = 18 - 6 = 12
    Number taking Physics and Biology only: ∣P∩B∣−∣P∩C∩B∣=15−6=9|P \cap B| - |P \cap C \cap B| = 15 - 6 = 9
    Number taking Chemistry and Biology only: ∣C∩B∣−∣P∩C∩B∣=12−6=6|C \cap B| - |P \cap C \cap B| = 12 - 6 = 6

  8. Sum these values to find the total number taking exactly two subjects.

    Exactly two subjects=12+9+6\text{Exactly two subjects} = 12 + 9 + 6

  9. Perform the calculation.

    Exactly two subjects=27\text{Exactly two subjects} = 27

    The number of students taking exactly two subjects is 27\boxed{\text{27}}.

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