Question

in a survey on 136 boys, 67 played golf, 56 played cricket and 40 played football, 11 played both golf and cricket, 12 played both cricket and football, 9 played both golf and football. each boy played at least one of these games. how many played all the three games.

Answer 1

Using the principle of inclusion and exclusion, the number of boys who played golf, cricket, and football is calculated to be 5.

Step-by-step explanation:

How Many Boys Played All Three Sports?

To figure out the number of boys who played golf, cricket, and football, we must use the principle of inclusion and exclusion. We start by adding the number of boys who played each sport: 67 played golf, 56 played cricket, and 40 played football. Then we subtract the numbers who played the sports in pairs to avoid double-counting: 11 played both golf and cricket, 12 played both cricket and football, 9 played both golf and football. However, we've now subtracted those who played all three sports twice, so we must add back this number once to get the correct total.

Let's define the number of boys who played all three sports as x. The sum of the individual sports is:

67 (golf) + 56 (cricket) + 40 (football) = 163.

The sum of the boys who played two sports is:

11 (golf and cricket) + 12 (cricket and football) + 9 (golf and football) = 32.

So, 163 (total single sports) - 32 (twice the number who played two sports) + x (those who played all three, added back) equals the total number of boys who played sports, 136.

Therefore, our equation is 163 - 32 + x = 136. Solving for x, we get x = 136 - 131 = 5 boys who played all three sports.

So the number of boys who played golf, cricket, and football is 5.