# Venn Diagram Question

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There are $150$ students in a class. The number of students who play Cricket, Hockey, and Basketball are $125, 130, 135$ respectively. If $5$ students do not play any of the three games, the number of students playing all the three games must be at least

1. $90$
2. $95$
3. $100$
4. $105$

100 should be the correct answer.

We have 125 Cricket Players, 130 Hockey Players & 135 Basketball Players.

Thus we have total (125 + 130 + 135) = 390 game players.

In the class we have 145 Player students & 5 non player students.

Each of these 145 Player students plays at least one of these 3 games( otherwise he/she would come in non player's category),

Now out of these 145 Players students we will have some 1 game players, some 2 game players and some 3 game players.

That is each of the player student can be either a 1 game players, or a 2 game player or a 3 game player.

Also all of these 145 players students will make up 390 game players.

Now suppose if all 145 students are 1 game players, then we will have 145 x 1 = 145 game players.

suppose if all 145 students are 2 game players, then we will have 145 x 2 = 290 game players,

suppose if all 145 students are 3 game players, then we will have 145 x 3 = 435 game players.

It is clear that to make  390 game players out of 145 player students, we need some 3 game player students also, because even if all of 145 students were 2 game players then also we can have only 145 x 2 = 290 players at most, and we are still 100 game players away from given 390 players hence we need at least (390 - 290) = 100, 3 game player students out of 145 player students using pigeon - hole principle.

(125 + 130 + 135) % 145 = 100

3.100

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