Question

25 students of your school participated in a tournament of three games namely : cricket, football & basket ball. 15 students received medals in cricket, 12 in football, 11 in basketball, 5 in cricket and basket ball, 9 in cricket & football, 4 in football & basketball and 3 in all the three games. How many students received medals in (i) None of the games (ii) cricket only.

Answer 1

Answer on Question #57573 – Math – Combinatorics | Number Theory

Question

25 students of your school participated in a tournament of three games namely : cricket, football & basket ball.

15 students received medals in cricket, 12 in football, 11 in basketball,

5 in cricket and basket ball, 9 in cricket & football, 4 in football & basketball

and 3 in all the three games.

How many students received medals in

(i) None of the games

(ii) (ii) cricket only.

Solution

Using this diagram

All games medalist: $|set5| = 3$

Only cricket & football: $|set3| = 9 - 3 = 6$

Only football & basketball: $|set7| = 4 - 3 = 1$

Only cricket & basketball: $|set6| = 5 - 3 = 2$

Only football: $|set4| = 12 - 6 - 1 - 3 = 2$

ii) Only cricket: $|set2| = 15 - 6 - 2 - 3 = 4$

Only basketball: $|set8| = 11 - 2 - 1 - 3 = 5$

i)

First method

Either of games medalists:

$$| s e t 2 | + | s e t 3 | + | s e t 4 | + | s e t 5 | + | s e t 6 | + | s e t 7 | + | s e t 8 | = 3 + 6 + 1 + 2 + 2 + 4 + 5 = 2 3$$

None of the games: |set1| =25-23=2

Second method

By inclusion-exclusion principle, the number of students received medals in either of games is given by

$$\begin{array}{l} | A \cup B \cup C | = | A | + | B | + | C | - | A \cap B | - | A \cap C | - | B \cap C | + | A \cap B \cap C | = \\ = 15 + 12 + 11 - 5 - 9 - 4 + 3 = 23. \end{array}$$

Hence the number of students received medals in none of the games is

$$| \overline{A \cup B \cup C} | = | X | - | A \cup B \cup C | = 25 - 23 = 2$$

Answer: (i) 2 ; (ii) 4.

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