In a group of 240 game lovers, 135 like cricket and 120 like football. By drawing a Venn-diagram, find:

(i) how many people like both the games?

(ii) How many people like only cricket?

Answer: (i) 15, (ii) 120

Solution:

Let U be the set of total students surveyed. Let C and F represent the set of students who liked to play Cricket and Football, respectively.

Given,

$n(U) = 240$

$n(C) = 135$

$n(F) = 120$

$n( \overline{ L \cup P})= 0$

To find:

(i) $n(L \cap P) = ?$

(ii) $n_o(C) = ?$

We know,

$n(U) = n(C) + n(F) - n(C \cap F) + n( \overline{C \cup F}) $

$or, 240 = 135 + 120 - n(C \cap F) + 0$

$or, 240 = 255 - n(C \cap F)$

$or, n(C \cap F) = 255 - 240$

$\therefore n(C \cap F) = 15$

And,

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

$or, n_o(C) = 135 - 15$

$\therefore n_o(C) = 120$

Hence,

The number of students who liked both the games were 15.

The number of students who liked only Cricket were 120.

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