assuming that boys and girls are equally probable find the number of families out of 1600 each having 5 children with 3 boys,5 girls,2or3 boys and all children of the same sex.

To find the number of families out of 1600 with different combinations of children, we can use the principles of combinatorics. Let's break it down step by step:

1. Families with exactly 3 boys and 2 girls:
- To have exactly 3 boys and 2 girls, we need to choose 3 out of 5 children to be boys (since there are 5 total children) and the remaining 2 will be girls.
- The number of ways to choose 3 boys out of 5 is given by the binomial coefficient "5 choose 3" (symbolized as 5C3) or "5 over 3". It can be calculated as:
5C3 = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4) / (2 * 1) = 10
- So, there are 10 ways to choose 3 boys and 2 girls in a family.

2. Families with exactly 5 boys and 0 girls (all boys):
- As per the given condition, we have 5 boys and 0 girls.
- Since there is only one possibility in this case (5 boys), the number of families is 1.

3. Families with exactly 0 boys and 5 girls (all girls):
- Similarly, when we have 0 boys and 5 girls, there is only one possibility (all girls).
- So, the number of families in this case is 1 as well.

4. Families with either 2 or 3 boys (with no girls):
- We need to consider families with either 2 or 3 boys but no girls.
- First, calculate the number of families with exactly 2 boys and no girls:
- Using the same logic as in point 1, 2 boys can be chosen out of 5 children (5C2) in 10 different ways.
- Next, calculate the number of families with exactly 3 boys and no girls:
- Using the same logic as before, 3 boys can be chosen out of 5 children (5C3) in 10 different ways.
- Therefore, the total number of families in this case is 10 + 10 = 20.

5. Families with all children of the same sex (either all boys or all girls):
- We need to consider families with either all boys or all girls (i.e., 5 boys and 0 girls, or 0 boys and 5 girls).
- As calculated in points 2 and 3, there is 1 family each with all boys or all girls.
- So, the total number of families in this case is 1 + 1 = 2.

Now, let's calculate the total number of families by adding up the numbers from each scenario:
10 (3 boys, 2 girls) + 1 (all boys) + 1 (all girls) + 20 (2 or 3 boys, no girls) + 2 (all boys or all girls) = 34

Therefore, there are 34 families out of 1600 with the specified combinations of children.