A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist of 1 man and 2 women?

Step 1: Calculate total ways to form a committee

We need to form a committee of 3 persons from a total group of $2 \text{ men} + 3 \text{ women} = 5 \text{ persons}$. The number of ways to choose $k$ items from a set of $n$ items is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. Here, $n=5$ and $k=3$.

Step 2: Compute total combinations

Calculating the combination, we find that there are 10 different ways to form a committee of 3 persons from the group of 5.

Step 3: Calculate ways to choose 1 man from 2

Now, we need to find the number of committees that consist of 1 man and 2 women. First, let's determine the number of ways to choose 1 man from the available 2 men.

Step 4: Compute combinations for men

There are 2 ways to choose 1 man from the 2 available men.

Step 5: Calculate ways to choose 2 women from 3

Next, we determine the number of ways to choose 2 women from the available 3 women.

Step 6: Compute combinations for women

There are 3 ways to choose 2 women from the 3 available women.

Step 7: Calculate ways for specific committee composition

To find the total number of committees consisting of 1 man and 2 women, we multiply the number of ways to choose the men by the number of ways to choose the women.

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