In a group of 50 people , 30 like to play cricket , 25 like to play football and 32 like play hockey . Assume that each one like to play atleast one of the three games . If 15 people like to play both cricket as well as football, 11 people like to play both football well as hockey and 18 like to play both football as as hockey, then
(i) how many like to play all the three games?
(ii) how many like to play only football ?
(iii) how many like to play only hockey ?

Answer

To solve the problem, we will use the principle of inclusion-exclusion. Let's denote the sets as follows:

- Let \( A \) be the set of people who like cricket.
- Let \( B \) be the set of people who like football.
- Let \( C \) be the set of people who like hockey.

From the question, we have the following information:

- \( n(A) = 30 \) (people who like cricket)
- \( n(B) = 25 \) (people who like football)
- \( n(C) = 32 \) (people who like hockey)
- \( n(A \cap B) = 15 \) (people who like both cricket and football)
- \( n(B \cap C) = 11 \) (people who like both football and hockey)
- \( n(A \cap C) = 18 \) (people who like both cricket and hockey)
- \( n(A \cup B \cup C) = 50 \) (total number of people)

We need to find:

(i) The number of people who like all three games, \( n(A \cap B \cap C) \).

(ii) The number of people who like only football.

(iii) The number of people who like only hockey.

### Step 1: Find \( n(A \cap B \cap C) \)

Using the principle of inclusion-exclusion:

\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \]

Substituting the values we have:

\[ 50 = 30 + 25 + 32 - 15 - 11 - 18 + n(A \cap B \cap C) \]

Calculating the right-hand side:

\[ 50 = 87 - 44 + n(A \cap B \cap C) \]
\[ 50 = 43 + n(A \cap B \cap C) \]

Now, solving for \( n(A \cap B \cap C) \):

\[ n(A \cap B \cap C) = 50 - 43 = 7 \]

### Step 2: Find the number of people who like only football

To find the number of people who like only football, we can use the formula:

\[ n(B \text{ only}) = n(B) - n(A \cap B) - n(B \cap C) + n(A \cap B \cap C) \]

Substituting the values we have:

\[ n(B \text{ only}) = 25 - 15 - 11 + 7 \]
\[ n(B \text{ only}) = 25 - 26 + 7 = 6 \]

### Step 3: Find the number of people who like only hockey

To find the number of people who like only hockey, we can use the formula:

\[ n(C \text{ only}) = n(C) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]

Substituting the values we have:

\[ n(C \text{ only}) = 32 - 18 - 11 + 7 \]
\[ n(C \text{ only}) = 32 - 29 + 7 = 10 \]

### Final Answers

(i) The number of people who like all three games: **7**

(ii) The number of people who like only football: **6**

(iii) The number of people who like only hockey: **10**