Question

In a village of 120 families,93 families use firewood for cooking,63 families use kerosene,45 families use cooking gas,45 families use firewood and kerosene,24 families use kerosene and cooking gas,27 families use cooking gas and firewood. find how many use firewood,kerosene and cooking gas.

Answer 1

From the data, we get
n ( F ∪ K ∪ C ) = 120
n (F) = 93
n (K) = 63
n (C) = 45
n (F ∩ K) = 45
n (K ∩ C) = 24
n (C ∩ F) = 27

We need to find n (C ∩ F ∩ K)

We already have the formula:
n (A ∪ B ∪ C ) = n (A)  + n (B) + n (C) - n (A ∩ B) - n (B ∩ C) - n (A ∩ C) + n (A ∩ B ∩ C)

Using this formula, we can easily find the answer.
Thus, 
n (F ∪ K ∪ C ) = n (F)  + n (K) + n (C) - n (F ∩ K) - n (K ∩ C) - n (F ∩ C) + n (F ∩ K ∩ C)

120 = 93 + 63 + 45 - 45 - 24 - 27 + n (F ∩ K ∩ C)
120 = 105 + n (F ∩ K ∩ C)
n (F ∩ K ∩ C) = 120 - 105
n (F ∩ K ∩ C) = 15

Thus, the number of families using firewood, kerosene and cooking gas is 15.