Form a group of 6 men and 4 women a committee of 4 persons is to be formed:
In how many different ways can it be done so that the committee has at least one woman?

Options

A. 210
B. 225
C. 195
D. 185

Answer

To solve the problem of forming a committee of 4 persons from a group of 6 men and 4 women, ensuring that there is at least one woman in the committee, we can follow these steps:

### Step 1: Calculate the total number of ways to form a committee of 4 persons without any restrictions.
We have a total of 10 people (6 men + 4 women). The number of ways to choose 4 persons from 10 is given by the combination formula:

\[ \text{Total ways} = \binom{10}{4} \]

Calculating this:

\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

### Step 2: Calculate the number of ways to form a committee of 4 persons with no women (all men).
If we want to find the number of committees that have no women, we can only select from the 6 men. The number of ways to choose 4 men from 6 is:

\[ \text{Ways with no women} = \binom{6}{4} \]

Calculating this:

\[ \binom{6}{4} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

### Step 3: Subtract the number of all-male committees from the total number of committees.
To find the number of committees that have at least one woman, we subtract the number of all-male committees from the total number of committees:

\[ \text{Committees with at least one woman} = \text{Total ways} - \text{Ways with no women} \]

Calculating this:

\[ \text{Committees with at least one woman} = 210 - 15 = 195 \]

### Final Answer:
Thus, the number of different ways to form a committee of 4 persons that includes at least one woman is **195**.

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