How many ways are there to select a group of five from 6 males and 6 females
a) with no restrictions
b) if the group must contain exactly 4 males and 1 female
Is your question asking for every combination that can be chosen?
(a) 12C5 = 792
(b) 6C4 * 6C1 = 90
Yes its asking for every combination that can be chosen
To answer these questions, we can use the concept of combinations, which is a way to calculate the number of ways to select a group from a larger set.
a) With no restrictions, we can select any combination of males and females. To calculate this, we can use the formula for combinations, which is given by:
nCr = n! / (r! * (n-r)!),
where n is the total number of items and r is the number of items to be selected.
In this case, we have 6 males and 6 females and we want to select a group of 5. Thus, we need to calculate 11C5.
11! / (5! * (11-5)!) = 11! / (5! * 6!) = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 462.
Therefore, there are 462 ways to select a group of five from 6 males and 6 females with no restrictions.
b) If the group must contain exactly 4 males and 1 female, we need to consider this constraint while calculating the combinations.
First, we need to choose 4 males from the 6 available. This can be calculated as 6C4.
Then, we need to choose 1 female from the 6 available. This can be calculated as 6C1.
Finally, we multiply these two results together to get the total number of combinations:
6C4 * 6C1 = (6! / (4! * 2!)) * (6! / (1! * 5!)) = (6 * 5 / (4 * 3 * 2 * 1)) * (6 / 1) = 15 * 6 = 90.
Therefore, there are 90 ways to select a group of five from 6 males and 6 females if the group must contain exactly 4 males and 1 female.