Question

По формуле бинома Ньютона Сколькими способами группу из 12 юношей и 8 девушек можно разбить на две группы по 10 человек так, чтобы в каждой из образовавшихся групп оказалось по 4 девушки ?

Answer 1

Решение во вложении
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Answer 2

Calculating the Number of Ways to Divide the Group

To calculate the number of ways to divide a group of 12 boys and 8 girls into two groups of 10 people each, with each group containing 4 girls, we can use the formula for combinations from the binomial theorem.

The formula for combinations is given by:

C(n, r) = n! / (r! * (n-r)!)

Where: - n is the total number of items - r is the number of items to choose - ! denotes factorial, which means the product of all positive integers up to that number

In this case, we want to choose 4 girls out of 8, and 6 boys out of 12 for each group. We can calculate the number of ways to do this using the combination formula.

Applying the Combination Formula

The number of ways to choose 4 girls out of 8 is given by: C(8, 4) = 8! / (4! * (8-4)!)

The number of ways to choose 6 boys out of 12 is given by: C(12, 6) = 12! / (6! * (12-6)!)

Calculating the Total Number of Ways

To find the total number of ways to divide the group into two groups of 10 with 4 girls in each, we can multiply the number of ways to choose the girls by the number of ways to choose the boys.

So, the total number of ways is: C(8, 4) * C(12, 6)

Final Calculation

Let's calculate the total number of ways using the given combination formula:

C(8, 4) = 8! / (4! * 4!) = 70 C(12, 6) = 12! / (6! * 6!) = 924

Therefore, the total number of ways to divide the group is: 70 * 924 = 64,680

So, there are 64,680 ways to divide the group of 12 boys and 8 girls into two groups of 10 people each, with each group containing 4 girls.