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?
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Step-by-Step Solution
Step 1: Calculate total ways to form a committee
We need to form a committee of 3 persons from a total group of 2 men+3 women=5 persons. The number of ways to choose k items from a set of n items is given by the combination formula (kn)=kn!!(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.