Answer to Question #250632 in Math for Kofi

Question

In a group of 120 students, 72 of them play football, 65 play table tennis and 53 play hockey. If 35 of the students play both football and table tennis, 30 play both football and hockey, 21 play table tennis and hockey and each of them plays at least one of the three games,

a) draw a Venn diagram to illustrate this information

b) how many of the students play:

i) all the three games

ii) exactly two of the three games iii) exactly one of the three games iv) football alone

Answer 1

a) Let $x=$ the number students playing all the three games.

The number of students playing exactly football and table tennis is $35-x.$

The number of students playing exactly football and hockey is $30-x.$

The number of students playing exactly table tennis and hockey is $21-x.$

The number of students playing only football is

$$72-(35-x)-(30-x)-x=x+7$$

The number of students playing only table tennis is

$$65-(35-x)-(21-x)-x=x+9$$

The number of students playing only table hocket is

$$53-(30-x)-(21-x)-x=x+2$$

Then the total number of students is

$$x+7+x+9+x+2+35-x$$

$$+30-x+21-x+x=120$$

$$x=16$$

b)

i)

$16$ students play all the three games.

ii)

$$35-16+30-16+21-16=38$$

$38$ students play exactly two of the three games.

iii)

$$16+7+16+9+16+2=66$$

$66$ students play exactly one of the three games .

iv)

$$16+7=23$$

$23$ students play football alone.