A committee of 3 members is to be formed out of 5 men and 2 women. Find the number of ways of selecting the committee if it is to consist of at least one women.

Answer

To solve the problem of forming a committee of 3 members from 5 men and 2 women, with the condition that at least one woman must be included, we can break it down into steps.

### Step 1: Understand the total combinations without restrictions
First, we calculate the total number of ways to form a committee of 3 members from 7 people (5 men + 2 women) without any restrictions. This can be calculated using the combination formula:

\[ \text{Total combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

where \( n \) is the total number of people and \( r \) is the number of people to choose.

\[ \text{Total combinations} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 2: Calculate the combinations with no women
Next, we calculate the number of ways to form a committee of 3 members that consists of only men (i.e., no women). This means we need to select all 3 members from the 5 men:

\[ \text{Combinations with no women} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Subtract the combinations with no women from the total
Now, 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:

\[ \text{Combinations with at least one woman} = \text{Total combinations} - \text{Combinations with no women} \]

\[ \text{Combinations with at least one woman} = 35 - 10 = 25 \]

### Final Answer
Thus, the number of ways to select a committee of 3 members that includes at least one woman is **25**.

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