A group of 5 boys and 4 girls is to be chosen from a class of 8 boys and 6 girls. In how many ways can this be done?
Answer Details
To solve this problem, we need to use the concept of combinations.
The number of ways to choose a group of r objects from a set of n distinct objects is given by the formula:
C(n,r) = n! / (r! * (n-r)!)
where n! denotes the factorial of n (i.e., the product of all positive integers up to and including n), and r! denotes the factorial of r.
In this problem, we want to choose a group of 5 boys and 4 girls from a class of 8 boys and 6 girls. We can do this in two steps:
Step 1: Choose the 5 boys from the 8 boys.
This can be done in C(8,5) ways.
Step 2: Choose the 4 girls from the 6 girls.
This can be done in C(6,4) ways.
The total number of ways to choose the group of 5 boys and 4 girls is the product of the number of ways in Step 1 and the number of ways in Step 2:
C(8,5) * C(6,4) = (8! / (5! * 3!)) * (6! / (4! * 2!))
= (8*7*6 / (3*2*1)) * (6*5 / (2*1))
= 56 * 15
= 840
Therefore, the number of ways to choose a group of 5 boys and 4 girls from a class of 8 boys and 6 girls is 840.