A teacher has 20 boys and 10 girls in her class. In how many way can she select 6 of the children to be in a play if she must have 3 boys and 3 girls?
there are 20C3 ways to pick the boys
and 10C3 ways to pick the girls.
So, 20C3 * 10C3 ways to choose
No permutations involved, just combinations.
To find the number of ways the teacher can select 3 boys and 3 girls from her class, we can use combinations.
The number of ways to select 3 boys out of 20 is denoted as C(20, 3). This can be calculated as:
C(20, 3) = 20! / (3! * (20 - 3)!) = 20! / (3! * 17!) = (20 * 19 * 18) / (3 * 2 * 1) = 1140.
Similarly, the number of ways to select 3 girls out of 10 is denoted as C(10, 3). This can be calculated as:
C(10, 3) = 10! / (3! * (10 - 3)!) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120.
To find the total number of ways to select 3 boys and 3 girls, we multiply the two values:
Total number of ways = C(20, 3) * C(10, 3) = 1140 * 120 = 136,800.
So, there are 136,800 ways the teacher can select 3 boys and 3 girls to be in the play.
To solve this problem, we can use the concept of combinations. The number of ways to select 3 boys out of 20 can be calculated using the formula for combinations:
C(n, k) = n! / (k! * (n - k)!)
where n is the total number of boys (20) and k is the number of boys we want to select (3). Applying the formula, we have:
C(20, 3) = 20! / (3! * (20 - 3)!)
Similarly, we can calculate the number of ways to select 3 girls out of 10 using combinations:
C(10, 3) = 10! / (3! * (10 - 3)!)
Since the teacher needs to select both 3 boys and 3 girls, we then multiply these two results together:
Total number of ways = C(20, 3) * C(10, 3)
Now, let's calculate each part step by step:
C(20, 3) = 20! / (3! * (20 - 3)!)
= 20! / (3! * 17!)
= (20 * 19 * 18) / (3 * 2 * 1)
= 1140
C(10, 3) = 10! / (3! * (10 - 3)!)
= 10! / (3! * 7!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120
Total number of ways = C(20, 3) * C(10, 3)
= 1140 * 120
= 136,800
Therefore, there are 136,800 ways in which the teacher can select 3 boys and 3 girls from her class of 20 boys and 10 girls for the play.