Question

Задача легкая, но меня вводит в ступор. 10 человек случайным образом рассаживаются на десятиместную скамейку. какова вероятность, что 2 определенных лица окажутся рядом? Объясните решение

Answer 1

надо выяснить для себя 2 вещи

1. сколько всего разных случаев посадки 10 людей на 10 мест. Ясно что это 10!.

2. сколько из этих случаев устроены так, что 2 заданных человека рядом. Перый из них может сесть на 10 мест 10 способами. если он на краю, то второй может сесть 1 спосбом (если край левый, то справа, а если правый - то слева). Если же место со 2 по 9, то второй человек может сеcть 2 способами - либо справа либо слева. Поэтому всего вариантов рассадок 2 людей рядом на 10 местах 2 + 8*2 = 18. И на каждый из этих вариантов посадки остальные 8 человек могут сесть 8! способами на оставшиеся 8 мест. Всего выходит 18*8! = 2*9!. Легко можно понять, что в этих рассадках не встречается повторов, то есть не надо делить ничего и прочее. 

3. ПО ОПРЕДЕЛЕНИЮ вероятность это отношение числа нужных вариантов к общему числу вариантов (какого-то события). То есть 2*9!/10! = 1/5;

Answer 2

Problem Analysis

The problem states that 10 people are randomly seated on a 10-seat bench, and we need to find the probability that two specific individuals will be seated next to each other. To solve this problem, we can use the concept of permutations and combinations.

Solution

To find the probability, we need to determine the total number of possible seating arrangements and the number of seating arrangements where the two specific individuals are seated next to each other.

Total Number of Seating Arrangements

Since there are 10 people and 10 seats, the total number of seating arrangements can be calculated using the factorial function. The factorial of a number n (denoted as n!) is the product of all positive integers less than or equal to n.

In this case, the total number of seating arrangements is 10!.

Number of Seating Arrangements with Two Specific Individuals Seated Next to Each Other

To calculate the number of seating arrangements where the two specific individuals are seated next to each other, we can treat them as a single entity. This reduces the problem to arranging 9 entities (8 individuals + 1 pair) in 9 seats.

The number of seating arrangements with the two specific individuals seated next to each other can be calculated as (9 - 1)! since we treat the pair as a single entity.

Calculating the Probability

To find the probability, we divide the number of seating arrangements with the two specific individuals seated next to each other by the total number of seating arrangements.

Let's calculate the probability:

Total number of seating arrangements = 10!
Number of seating arrangements with two specific individuals seated next to each other = (9 - 1)!
Probability = (Number of seating arrangements with two specific individuals seated next to each other) / (Total number of seating arrangements)

Calculation

Let's calculate the probability using the formula:

Total number of seating arrangements = 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800
Number of seating arrangements with two specific individuals seated next to each other = (9 - 1)! = 8!
Probability = (8!) / (10!) = 8! / 10!

Now, we can calculate the probability using the formula:

Probability = (8!) / (10!) = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)

Simplifying the expression:

Probability = (8 * 7) / (10 * 9) = 56 / 90 = 28 / 45

Therefore, the probability that the two specific individuals will be seated next to each other is 28/45.

Conclusion

The probability that the two specific individuals will be seated next to each other on the 10-seat bench is 28/45.