In a class of 80 girls, 17 play football, 12 play hockey and 4 play both games. Find the number of girls who…

Question

In a class of 80 girls, 17 play football, 12 play hockey and 4 play both games. Find the number of girls who play i) Only one of the two games ii) None of the two games

ScanSolve AI Tutor · Accepted Answer

Here's how to solve this problem using set theory:

Define the sets and given values.
Let $U$ be the set of all girls in the class. $|U| = 30$ .
Let $F$ be the set of girls who play football. $|F| = 17$ .
Let $H$ be the set of girls who play hockey. $|H| = 12$ .
Let $F \cap H$ be the set of girls who play both games. $|F \cap H| = 4$ .
Calculate the number of girls who play only Football.
This is the number of girls in set $F$ but not in set $H$ .
$|F \text{ only}| = |F| - |F \cap H|$ $|F \text{ only}| = 17 - 4 = 13$
Calculate the number of girls who play only Hockey.
This is the number of girls in set $H$ but not in set $F$ .
$|H \text{ only}| = |H| - |F \cap H|$ $|H \text{ only}| = 12 - 4 = 8$
Calculate the number of girls who play only one of the two games (Part i).
This is the sum of girls who play only Football and girls who play only Hockey.
$|\text{Only one game}| = |F \text{ only}| + |H \text{ only}|$ $|\text{Only one game}| = 13 + 8 = 21$
The number of girls who play only one of the two games is $\boxed{\text{21}}$ .
Calculate the number of girls who play at least one of the two games.
This is the union of the two sets, $|F \cup H|$ .
$|F \cup H| = |F \text{ only}| + |H \text{ only}| + |F \cap H|$ $|F \cup H| = 13 + 8 + 4 = 25$
Alternatively, using the inclusion-exclusion principle:
$|F \cup H| = |F| + |H| - |F \cap H|$ $|F \cup H| = 17 + 12 - 4 = 29 - 4 = 25$
Calculate the number of girls who play none of the two games (Part ii).
This is the total number of girls minus those who play at least one game.
$|\text{None of the games}| = |U| - |F \cup H|$ $|\text{None of the games}| = 30 - 25 = 5$
The number of girls who play none of the two games is $\boxed{\text{5}}$ .

In a class of 80 girls, 17 play football, 12 play hockey and 4 play both games. Find the number of girls who play i) Only one of the two games ii) None of the two games

Step-by-step solution

Define the sets and given values.

Calculate the number of girls who play only Football.

Calculate the number of girls who play only Hockey.

Calculate the number of girls who play only one of the two games (Part i).

Calculate the number of girls who play at least one of the two games.

Calculate the number of girls who play none of the two games (Part ii).

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