Question

in a class of 26 students 14 have cats 10 have dogs and 5 have parrots.if 4 have dogs and cats,1 has dog and parrot ,3 have parrots and cats and no one has all the three.how many have none?​

Answer 1

Answer:  The number of students having none of the pets is 5.

Step-by-step explanation:  Given that in a class of 26 students, 14 have cats 10 have dogs, 5 have parrots, 4 have dogs and cats,1 has dog and parrot ,3 have parrots and cats and no one has all the three.

We are to find the number of students that have none of the three pets.

Let A, B and C represents the set of students having cats, dogs and parrots respectively.

Then, from the given information, we have

$n(A)=14,~~n(B)=10,~~n(C)=5,~~n(A\cap B)=4,~~n(B\cap C)=1,~~n(A\cap C)=3,\\\\n(A\cap B\cap C)=0.$

From set theory, the number of students having one of at least one of the pets is given by

$n(A\cup B\cup C)\\\\=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)\\\\=14+10+5-4-1-3+0\\\\=29-8\\\\=21.$

Therefore, the number of students having none of the pets is given by

$26-n(A\cup B\cap C)=26-21=5.$

Thus, the number of students having none of the pets is 5.