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In a group of 100 students, 60 play football, 50 play volleyball, and 40 play basketball. 25 play football and volleyball, 20 play football and basketball, 15 play volleyball and basketball, and 10 pl

Mathematics

ScanSolve AI Answer

Answer

25\text{25}

30\text{30}

0\text{0}

Step-by-step solution

10 steps

Alright selorm, let's tackle .

Let FF be the set of students who play Football, VV for Volleyball, and BB for Basketball.
Given:
Total students = 100
∣F∣=60|F| = 60
∣V∣=50|V| = 50
∣B∣=40|B| = 40
∣F∩V∣=25|F \cap V| = 25
∣F∩B∣=20|F \cap B| = 20
∣V∩B∣=15|V \cap B| = 15
∣F∩V∩B∣=10|F \cap V \cap B| = 10

1. Play only football
This means students who play Football but not Volleyball and not Basketball.

  1. Use the formula for "only F".

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

  2. Substitute the given values.

    ∣F only∣=60−(25+20)+10|F \text{ only}| = 60 - (25 + 20) + 10

  3. Perform the calculations.

    ∣F only∣=60−45+10|F \text{ only}| = 60 - 45 + 10 ∣F only∣=15+10|F \text{ only}| = 15 + 10 ∣F only∣=25|F \text{ only}| = 25

    The number of students who play only football is 25\boxed{\text{25}}.

    2. Play exactly two games
    This is the sum of students playing (Football and Volleyball only), (Football and Basketball only), and (Volleyball and Basketball only).

  4. Calculate the number of students playing exactly two games for each pair.

    Number playing Football and Volleyball only: ∣F∩V∣−∣F∩V∩B∣=25−10=15|F \cap V| - |F \cap V \cap B| = 25 - 10 = 15
    Number playing Football and Basketball only: ∣F∩B∣−∣F∩V∩B∣=20−10=10|F \cap B| - |F \cap V \cap B| = 20 - 10 = 10
    Number playing Volleyball and Basketball only: ∣V∩B∣−∣F∩V∩B∣=15−10=5|V \cap B| - |F \cap V \cap B| = 15 - 10 = 5

  5. Sum these values to find the total number playing exactly two games.

    Exactly two games=15+10+5\text{Exactly two games} = 15 + 10 + 5

  6. Perform the calculation.

    Exactly two games=30\text{Exactly two games} = 30

    The number of students who play exactly two games is 30\boxed{\text{30}}.

    3. Play none
    First, we find the number of students who play at least one game using the Principle of Inclusion-Exclusion.

  7. Apply the Principle of Inclusion-Exclusion formula to find ∣F∪V∪B∣|F \cup V \cup B|.

    ∣F∪V∪B∣=∣F∣+∣V∣+∣B∣−(∣F∩V∣+∣F∩B∣+∣V∩B∣)+∣F∩V∩B∣|F \cup V \cup B| = |F| + |V| + |B| - (|F \cap V| + |F \cap B| + |V \cap B|) + |F \cap V \cap B|

  8. Substitute the given values.

    ∣F∪V∪B∣=60+50+40−(25+20+15)+10|F \cup V \cup B| = 60 + 50 + 40 - (25 + 20 + 15) + 10

  9. Perform the calculations.

    ∣F∪V∪B∣=150−60+10|F \cup V \cup B| = 150 - 60 + 10 ∣F∪V∪B∣=90+10|F \cup V \cup B| = 90 + 10 ∣F∪V∪B∣=100|F \cup V \cup B| = 100

  10. Subtract the number of students playing at least one game from the total number of students.

    Number who play none=Total students−∣F∪V∪B∣\text{Number who play none} = \text{Total students} - |F \cup V \cup B| Number who play none=100−100\text{Number who play none} = 100 - 100 Number who play none=0\text{Number who play none} = 0

    The number of students who play none is 0\boxed{\text{0}}.

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