Each student in a class of 32 plays at least one game: cricket, football, or basketball. 20 play cricket, 18 play basketball, and 25 play football. 9 play cricket and basketball, 13 play football and basketball and 5 play all three. Find the number of students who play:

1. Cricket and football
2. Cricket and football but not basketball
3. Show the information in a Venn-diagram.

Solution:

Let U be the set of total students in the class.

Let C,F and B represent the sets of students who play Cricket, Football and Basketball, respectively.

Here,

$n(U) = 32$

$n(C) = 20$

$n(B) = 18$

$n(F) = 25$

$n(C \cap B) = 9$

$n(F \cap B) = 13$

$n(C \cap F \cap B) = 5$

Using formula,

$n(U) = n(C) + n(B) + n(F) - n(C \cap B ) - n(F \cap B) - n(C \cap F) + n(C \cap F \cap B)$

$or, 32 = 20 + 18 + 25 - 9 - 13 - n(C \cap F) + 5$

$or, 32 = 46 - n(C \cap F)$

$\therefore n(C \cap F) = 14$

So, 14 students play both cricket and football.

Now,

$n_o(C \cap F) = n(C \cap F) -n(C \cap F \cap B)$

$= 14 - 5$

$= 9$

Hence, 9 students liked to play football and cricket but not basketball.

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