Question

each student in a class of 35 plays at least one game among chess carrom and table tennis 22 play chess 21 play carrom 15play table tennis 10play chess and table tennis 8 play carrom and table tennis and 6 play all the three games? find the number of students who play chess and carrom but not table tennis (ii) only chess (iii) only carrom use venn diagram

Answer 1

Answer: (i) The students who play chess and carrom but not table tennis is 5. (ii) Students who plays only chess is 7. (iii) Students who plays only carrom is 8.

Explanation:

It is given that each student in a class of 35 plays at least one game among chess carrom and table tennis. 22 play chess 21 play carrom 15 play table tennis 10 play chess and table tennis 8 play carrom and table tennis and 6 play all the three games.

Let student who plays chess is denoted by A, student who plays carrom is denoted by B and student who plays table tennis is denoted by C.  

From given information A=22, B=21, C=15, $A\cap C=10$, $B\cap C=8$ and $A\cap B\cap C=6$

The student who plays chess or carrom but not table tennis is

$B\cap C\cap A'=B\cap C-A\cap B\cap C=8-6=2$

$A\cap C\cap B'=A\cap C-A\cap B\cap C=10-6=4$

$A'\cap B'\cap C=C-(A'\cap B\cap C)-(A\cap B'\cap C)-(A\cap B\cap C)\\A'\cap B'\cap C=15-2-4-6=3$

$A\cap B=A+B-A\cup B\\A\cap B=22+21-32=11$

$A\cup B\cup C'=A\cap B-A\cap B\cap C\\A\cup B\cup C'=11-6=5$

$A\cap B'\cap C'=A-(A\cap B'\cap C)-(A\cap B\cap C')-(A\cap B\cap C)\\A'\cap B'\cap C=22-4-5-6=7$

$A'\cap B\cap C'=B-(A\cap B\cap C')-(A'\cap B\cap C)-(A\cap B\cap C)\\A'\cap B'\cap C=21-5-2-6=8$

(i) We have to find the students who play chess and carrom but not table tennis, so we have to find $A\cup B\cup C'$. So the students who play chess and carrom but not table tennis is 5.

(ii) We have to find the students who play only chess , so we have to find $A\cap B'\cap C'$. So the students who play who play only chess is 7.

(iii)We have to find the students who play only carrom , so we have to find $A'\cap B\cap C'$. So the students who play who play only carrom is 8.