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

The Correct Option is C

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.