From a group of 4 boys and 3 girls, candidates are arranged at random, one after the other, for an interview. The probability that the boys and girls alternate, is

Options

A. `1//34`
B. `1//35`
C. `1//33`
D. `1//32`

Answer

To solve the problem of finding the probability that boys and girls alternate when arranging candidates for an interview from a group of 4 boys and 3 girls, we can follow these steps:

### Step 1: Calculate the Total Arrangements
The total number of candidates is 4 boys + 3 girls = 7 candidates. The total arrangements of these 7 candidates can be calculated using the factorial function:

\[ \text{Total arrangements} = 7! = 5040 \]

### Step 2: Calculate the Arrangements with Alternating Boys and Girls
To have an alternating arrangement, we can start with either a boy or a girl. However, since we have more boys than girls, we will start with a boy.

The arrangement will look like this: B G B G B G B

- There are 4 boys and 3 girls.
- The boys can be arranged among themselves in \(4!\) ways.
- The girls can be arranged among themselves in \(3!\) ways.

Thus, the number of arrangements where boys and girls alternate is:

\[ \text{Alternating arrangements} = 4! \times 3! = 24 \times 6 = 144 \]

### Step 3: Calculate the Probability
The probability that the boys and girls alternate is given by the ratio of the number of favorable arrangements to the total arrangements:

\[ \text{Probability} = \frac{\text{Alternating arrangements}}{\text{Total arrangements}} = \frac{144}{5040} \]

### Step 4: Simplify the Probability
To simplify \(\frac{144}{5040}\):

\[ \frac{144}{5040} = \frac{1}{35} \]

### Final Answer
Thus, the probability that the boys and girls alternate is:

\[ \frac{1}{35} \]