Question

Каким числом способов можно из 13 различных пар ботинком выбрать 8 ботинок так,чтобы все они были на одну ногу

Answer 1

13 пар ботинок ⇒⇒13 ботинок на левую ногу и 13 на правую.
Количество сочетаний из 13 по 8 равно

$x_{13} ^{8} = \frac{13!}{8!(13-8)!} = \frac{13*12*11*10*9*(8!)}{(8!)*5*4*3*2*1} =1 287$

Значит,способов выбрать 8 ботинок на одну ногу 1 287,а на другую столько же ⇒⇒умножим на 2
1287*2=2 574 способа

Answer 2

Calculation of the Number of Ways to Select 8 Boots from 13 Different Pairs

To calculate the number of ways to select 8 boots from 13 different pairs such that all of them are on one leg, we can break down the problem into two steps:

1. Selecting 4 pairs of boots: Since all the boots need to be on one leg, we need to select 4 pairs of boots out of the 13 available pairs. The order of selection does not matter.

2. Selecting one boot from each of the 4 selected pairs: Once we have selected the 4 pairs, we need to choose one boot from each of these pairs. Again, the order of selection does not matter.

To calculate the number of ways for each step, we can use the concept of combinations.

Step 1: Selecting 4 pairs of boots The number of ways to select 4 pairs out of 13 is given by the combination formula:

C(13, 4) = 13! / (4! * (13 - 4)!)

Step 2: Selecting one boot from each of the 4 selected pairs For each of the 4 selected pairs, we need to choose one boot. Since each pair has 2 boots, the number of ways to select one boot from each pair is 2^4.

Therefore, the total number of ways to select 8 boots from 13 different pairs such that all of them are on one leg is:

C(13, 4) * 2^4

Let's calculate this value:

C(13, 4) = 13! / (4! * (13 - 4)!) = 715

2^4 = 16

Therefore, the total number of ways to select 8 boots from 13 different pairs such that all of them are on one leg is:

715 * 16 = 11,440

So, there are 11,440 ways to select 8 boots from 13 different pairs such that all of them are on one leg.