From 4 children, 2 women and 4 men , 4 are selected. Probability that there are exactly 2 children among the selected is

Options

A. `(2)/(7)`
B. `(3)/(7)`
C. `(10)/(21)`
D. `(2)/(10)`

Answer

To solve the problem of finding the probability that exactly 2 children are selected from a group of 4 children, 2 women, and 4 men when selecting 4 individuals, we can follow these steps:

### Step 1: Determine the total number of people
We have:
- 4 children
- 2 women
- 4 men

Total number of people = 4 (children) + 2 (women) + 4 (men) = **10 people**.

### Step 2: Calculate the total ways to select 4 people from 10
The total number of ways to select 4 people from 10 can be calculated using the combination formula:
\[ \text{Total ways} = \binom{10}{4} \]

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

### Step 3: Calculate the number of favorable outcomes
We need to select exactly 2 children from the 4 available, and then we need to select the remaining 2 individuals from the other 6 people (2 women + 4 men).

1. **Select 2 children from 4**:
\[ \text{Ways to select 2 children} = \binom{4}{2} \]
Calculating \(\binom{4}{2}\):
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

2. **Select 2 people from the remaining 6 (2 women + 4 men)**:
\[ \text{Ways to select 2 from 6} = \binom{6}{2} \]
Calculating \(\binom{6}{2}\):
\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

3. **Total favorable outcomes**:
\[ \text{Favorable outcomes} = \binom{4}{2} \times \binom{6}{2} = 6 \times 15 = 90 \]

### Step 4: Calculate the probability
The probability of selecting exactly 2 children is given by the ratio of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{90}{210} \]

### Step 5: Simplify the probability
To simplify \(\frac{90}{210}\):
\[ \frac{90 \div 30}{210 \div 30} = \frac{3}{7} \]

### Final Answer
The probability that exactly 2 children are selected is \(\frac{3}{7}\).

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