In a group photo, 6 individuals are to be arranged in two rows. The first row will have 2 people, and the second row will have 4. Among the group are Anna and Ben. If the positions are randomly assigned, what is the probability that Anna is placed in the rightmost spot of the front row, while Ben is positioned somewhere in the back row? Express your answer in terms of factorials.

Answer Choices

Correct answer marked below

A

\frac{4! \times 4!}{6!}

B

\frac{4! \times 3!}{6!}

\frac{1 \times 4 \times 4!}{6!}

Correct

D

\frac{1 \times 4!}{6!}

E

\frac{2 \times 4!}{6!}

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Official Explanation

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To solve the problem, we need to determine the probability that Anna is placed in the rightmost spot of the front row while Ben is positioned somewhere in the back row. Here's a step-by-step explanation:

Step 1: Total Number of Arrangements

First, we calculate the total number of ways to arrange the 6 individuals in two rows:

- Front row: 2 spots

- Back row: 4 spots

The total number of ways to arrange 6 people in these spots is the number of permutations of 6 distinct positions:

\text{Total arrangements} = 6!

Step 2: Fix Anna in the Rightmost Front Spot

We want Anna to be in the rightmost spot of the front row. This fixes Anna's position:

- Front row: [Any person, Anna]

The number of ways to arrange the remaining 5 people (excluding Anna) in the remaining 5 spots is:

5!

However, since Anna's position is fixed, we don't need to consider her in the permutation of the remaining spots.

Step 3: Ensure Ben is in the Back Row

Now, we need Ben to be in one of the 4 back row spots.

- Back row: 4 spots

- Front row: 1 spot remaining (since Anna occupies the rightmost front spot)

The probability that Ben is placed in the back row is the number of back row spots divided by the remaining spots after placing Anna:

\frac{4}{5}

Step 4: Calculate the Desired Arrangements

The number of favorable arrangements is the number of ways to place Ben in the back row multiplied by the number of ways to arrange the remaining 4 people (excluding Anna and Ben) in the remaining 4 spots:…

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