First slide

Operations on Sets

Question

Of the number of three athletic teams in a school, $21$ are in the basketball team, $26$ in hockey team and $29$ in the football team, $14$ play hockey and basketball, $15$ play hockey and football, $12$ play football and basketball and $8$ play all the games. The total number of members is

Moderate

A

$42$
B

$43$
C

$45$
D

none of these

Solution

Let $B, H, F$ be the sets of the three teams respectively
so $n (B) = 21, n (H) = 26, n (F) = 29,$
$\begin{array}{l} n (H \cap B) = 14 n (H \cap F) = 15, n (F \cap B) = 12 \\ n (B \cap H \cap F) = 8 \end{array}$
and
$\begin{array}{l} n (B \cup H \cup F) = n (B) + n (H) + n (F) - n (H \cap B) - n (H \cap F) - n (F \cap B) + n (B \cap H \cap F) \\ = 21 + 26 + 29 - 14 - 15 - 12 + 8 = 43 . \end{array}$

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