Question

۱۵) پدر و مادری سه فرزند دارند، در یک عکس یادگاری با کدام احتمال پدر در سمت راست عکس، مادر در سمت چپ عکس و فرزندان بین آن ها قرار دارند؟

Explanation:

⚡ Using: theoretical probability · 🆕 New: Permutations and Arrangements
Arranging items in a specific order

Step 1: Find the total number of possible arrangements

There are 5 people in total: a father, a mother, and 3 children.

The total number of ways to arrange 5 distinct people in a line for a photo is given by the permutation of 5 elements:

$$ n(S) = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $$

Step 2: Find the number of favorable arrangements

We want to find the number of arrangements where:

• The father is on the far right of the photo.
• The mother is on the far left of the photo.
• The 3 children are positioned between them.

This fixes the positions of the father and the mother:

• Position 1 (far left): Mother (1 choice)
• Position 5 (far right): Father (1 choice)

The remaining 3 positions in the middle must be filled by the 3 children. The number of ways to arrange 3 children in these 3 positions is:

$$ n(A) = 3! = 3 \times 2 \times 1 = 6 $$

Step 3: Calculate the probability

The probability \( P(A) \) of this specific arrangement occurring is the ratio of the number of favorable outcomes to the total number of outcomes:

$$ P(A) = \frac{n(A)}{n(S)} = \frac{6}{120} = \frac{1}{20} $$

Answer:

$$ \frac{1}{20} $$