Question

In a group of 160 pupils, 46 pupils failed Mathematics, 52 pupils failed History and 50 pupils failed Geography;31 pupils failed Mathematics and History, 33 failed History And Geography, 36 failed Mathematics and Geography and 24 failed all three subjects. find the number of pupils who failed atleast one subject.

Answer 1

Answer:

" 72 pupils failed in at least one subject. "

Solution:

Let, A := pupils failing in Mathematics

⇒n(A) = 46,

B := pupils failing in History

⇒ n(B) = 52 and

C := pupils in geography

⇒n(C) = 50.

Then, A ∩ B := pupils failing in both Mathematics and History

⇒n(A ∩ B) = 31,

B ∩ C := pupils failing in both History and geography

⇒n(B ∩ C) = 33 and

C n A := pupils failing in both geography and Mathematics

⇒n(C ∩ A) = 36.

Also, (A ∩ B ∩ C) := pupils failing in all the three subjects

⇒n(A ∩ B ∩ C) = 24.

We have to find the number of pupils who failed in atleast one subject, which is defined by (A ∪ B∪ C).

Now, n(A ∪ B ∪ C)

= n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C)

= 46 + 52 + 50 - 31 - 33 - 36 + 24

= 72

∴ 72 pupils failed in atleast one subject.