Out of 1000 boys in a college, 220 played cricket, 250 played hockey and 350 played basketball. Of the total 80 played both basketball and hockey, 100 played cricket and basketball and 50 played cricket and hockey, 30 played all three games. The number of boys who play atleast one game is :

Options

A. 500
B. 590
C. 600
D. 620

Answer

To find the number of boys who play at least one game, we can use the principle of inclusion-exclusion. Let's denote:

- \( N_C \): Number of boys who played cricket = 220
- \( N_H \): Number of boys who played hockey = 250
- \( N_B \): Number of boys who played basketball = 350
- \( N_{C \cap H} \): Number of boys who played both cricket and hockey = 50
- \( N_{C \cap B} \): Number of boys who played both cricket and basketball = 100
- \( N_{H \cap B} \): Number of boys who played both hockey and basketball = 80
- \( N_{C \cap H \cap B} \): Number of boys who played all three games = 30

We want to find the total number of boys who play at least one game, which is given by the formula:

\[ N_{C \cup H \cup B} = N_C + N_H + N_B - N_{C \cap H} - N_{C \cap B} - N_{H \cap B} + N_{C \cap H \cap B} \]

Now, substituting the values into the formula:

\[ N_{C \cup H \cup B} = 220 + 250 + 350 - 50 - 100 - 80 + 30 \]

Calculating step by step:

1. Calculate the sum of boys who played each game:
\[ 220 + 250 + 350 = 820 \]

2. Calculate the sum of boys who played two games:
\[ 50 + 100 + 80 = 230 \]

3. Now, substitute these results back into the formula:
\[ N_{C \cup H \cup B} = 820 - 230 + 30 \]

4. Calculate \( 820 - 230 \):
\[ 820 - 230 = 590 \]

5. Finally, add 30:
\[ 590 + 30 = 620 \]

Thus, the number of boys who play at least one game is \( \boxed{620} \).