A committee of 3 members is to be formed out of 3 men and 4 women. In how many different ways can it be done so that at least one number is a women?

Options

A. 34
B. 12
C. 30
D. 36

Answer

To solve the problem of forming a committee of 3 members from 3 men and 4 women, ensuring that at least one member is a woman, we can use a combinatorial approach.

### Step-by-Step Solution:

1. **Calculate the Total Ways to Form a Committee Without Restrictions:**
We can select any 3 members from the total of 7 people (3 men + 4 women). The total number of ways to choose 3 members from 7 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 7 \) and \( r = 3 \):
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

2. **Calculate the Ways to Form a Committee with No Women:**
This means we only select from the 3 men. The number of ways to choose 3 men from 3 is:
\[ \binom{3}{3} = 1 \]

3. **Calculate the Ways to Form a Committee with At Least One Woman:**
To find the number of ways to form a committee with at least one woman, we subtract the number of all-male committees from the total number of committees:
\[ \text{Ways with at least one woman} = \text{Total ways} - \text{Ways with no women} \]
Substituting the values we calculated:
\[ \text{Ways with at least one woman} = 35 - 1 = 34 \]

Thus, the total number of different ways to form a committee of 3 members such that at least one member is a woman is **34**.