Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey ; 80 played cricket and basketball and 40 played cricket and hockey 24 player all the three games. The number of boys who did not play any game is

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion to find the number of boys who played at least one game and then subtract that from the total number of boys to find those who did not play any game.

### Step 1: Define the Variables
Let:
- \( N = 800 \) (total number of boys)
- \( n_C = 224 \) (boys who played cricket)
- \( n_H = 240 \) (boys who played hockey)
- \( n_B = 336 \) (boys who played basketball)
- \( n_{H \cap B} = 64 \) (boys who played both hockey and basketball)
- \( n_{C \cap B} = 80 \) (boys who played both cricket and basketball)
- \( n_{C \cap H} = 40 \) (boys who played both cricket and hockey)
- \( n_{C \cap H \cap B} = 24 \) (boys who played all three games)

### Step 2: Apply the Inclusion-Exclusion Principle
The formula for the number of boys who played at least one game is given by:
\[ n_{C \cup H \cup B} = n_C + n_H + n_B - n_{C \cap H} - n_{C \cap B} - n_{H \cap B} + n_{C \cap H \cap B} \]

### Step 3: Substitute the Values
Substituting the known values into the formula:
\[ n_{C \cup H \cup B} = 224 + 240 + 336 - 40 - 80 - 64 + 24 \]

### Step 4: Calculate the Total
Now, we perform the calculations step by step:
1. Sum of individual games:
\[ 224 + 240 + 336 = 800 \]
2. Sum of those who played two games:
\[ 40 + 80 + 64 = 184 \]
3. Adding those who played all three games:
\[ 800 - 184 + 24 = 640 \]

Thus, the number of boys who played at least one game is:
\[ n_{C \cup H \cup B} = 640 \]

### Step 5: Calculate the Number of Boys Who Did Not Play Any Game
To find the number of boys who did not play any game, we subtract the number of boys who played at least one game from the total number of boys:
\[ \text{Boys who did not play any game} = N - n_{C \cup H \cup B} = 800 - 640 = 160 \]

### Final Answer
The number of boys who did not play any game is **160**.