a class of 15 student with 5 girls and 10 boys.they are divided into 3 groups each of 5 to do different group projectA,B,C respectively .how many ways can be grouped (A)if no restriction?(B)each group has only one gender?
A) To find the number of ways the students can be grouped for project A without any restriction, we can use the concept of combinations.
In this case, we have 15 students and we need to form groups of 5 for project A. The order of the groups does not matter.
The formula for combinations is given by:
C(n, r) = n! / (r!(n-r)!)
Where:
- n is the total number of students (15 in this case)
- r is the number of students per group (5 in this case)
- ! represents the factorial operation
Using this formula, we can calculate the number of ways to form groups without any restriction:
C(15, 5) = 15! / (5!(15-5)!)
Simplifying the equation:
C(15, 5) = 15! / (5! * 10!)
Calculating the factorial terms:
C(15, 5) = (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1)
After simplification, we get:
C(15, 5) = 3003
Therefore, there are 3003 ways to group the students for project A without any restriction.
B) If each group is required to have only one gender, we need to consider the different possibilities of having all girls or all boys in a group.
For the first group, we can select 5 girls from the 5 available girls, which can be calculated as C(5, 5) = 1.
For the second group, we can select 5 boys from the 10 available boys, which can be calculated as C(10, 5) = 252.
Finally, for the third group, we can select any gender since all remaining students will have the same gender as each other after selecting the first two groups.
Therefore, the total number of ways to group the students for project A, given the restriction of each group having only one gender, is:
Total number of ways = Number of ways for the first group * Number of ways for the second group * Number of ways for the third group
= 1 * 252 * 1
= 252
Hence, there are 252 ways to group the students for project A, with each group having only one gender.