Out if five girls and three boys, four children are to be randomly selected for contest . What is the probability that all the selected children are girls ?

Options

A. `(1)/(14)`
B. `(1)/(7)`
C. `(5)/(17)`
D. `(2)/(17)`

Explanation

To find the probability that all selected children are girls from a group of 5 girls and 3 boys, we can follow these steps:

### Step 1: Determine the total number of ways to select 4 children from 8.
We have a total of 8 children (5 girls and 3 boys). The number of ways to select 4 children from 8 can be calculated using the combination formula:

\[ \text{Total ways} = \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

Calculating this gives:

\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

### Step 2: Determine the number of ways to select 4 girls from 5.
Next, we need to find the number of ways to select 4 girls from the 5 available. This is also calculated using the combination formula:

\[ \text{Ways to select girls} = \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} \]

Calculating this gives:

\[ \binom{5}{4} = 5 \]

### Step 3: Calculate the probability.
Now, we can find the probability that all selected children are girls by dividing the number of ways to select 4 girls by the total number of ways to select 4 children:

\[ \text{Probability} = \frac{\text{Ways to select girls}}{\text{Total ways}} = \frac{5}{70} \]

### Step 4: Simplify the probability.
We can simplify \(\frac{5}{70}\):

\[ \frac{5}{70} = \frac{1}{14} \]

### Final Answer:
The probability that all selected children are girls is \(\frac{1}{14}\).

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