Question

In a group of 15 women, 7 have nose studs, 8 have ear rings and 3 have neither. How many of these have both nose studs and ear rings?

• A. 0
• B. 2
• C. 3
• D. 7

Answer 1

Answer: (C)

3

Answer 2

The correct option is (C): 3
To solve the problem, we can use the principle of inclusion-exclusion. Let:

- $A$: the set of women with nose studs
- $B$: the set of women with ear rings

Given:
- $|A| = 7$ (women with nose studs)
- $|B| = 8$ (women with ear rings)
- $|A' \cap B'| = 3$ (women with neither)

First, we calculate the total number of women who have either nose studs or ear rings (or both):

$|A \cup B| = \text{Total Women} - \text{Neither} = 15 - 3 = 12$

Now, we use the formula for the union of two sets:

$|A \cup B| = |A| + |B| - |A \cap B|$

Substituting the known values:

$12 = 7 + 8 - |A \cap B|$

This simplifies to:

$12 = 15 - |A \cap B|$

So,

$|A \cap B| = 15 - 12 = 3$

Therefore, the number of women who have both nose studs and ear rings is 3.