How many ways are there to split 12 people into 4 groups with 3 people in each group?
12C3 * 9C3 * 6C3 * 3C3
In how many ways can you group 12 people into 2 groups of 3 people and 3 groups of 2 people?
To find the number of ways to split 12 people into 4 groups, with 3 people in each group, you can use a combination formula.
First, let's find the number of ways to select 3 people from 12 to form the first group. This can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of elements (12 in this case) and r is the number of elements selected (3 in this case).
In this scenario, we are selecting 3 people from 12, so:
C(12, 3) = 12! / (3!(12-3)!) = 12! / (3!9!) = (12x11x10) / (3x2x1) = 220
Next, after selecting the first group, we have 9 people remaining and need to select 3 for the second group. Again, using the combination formula:
C(9, 3) = 9! / (3!(9-3)!) = 9! / (3!6!) = (9x8x7) / (3x2x1) = 84
Following the same logic, we need to select 3 people from the remaining 6 for the third group:
C(6, 3) = 6! / (3!(6-3)!) = 6! / (3!3!) = (6x5x4) / (3x2x1) = 20
Finally, we have 3 people left for the last group, and there is only one way to assign them to the final group.
So, to find the total number of ways to split 12 people into 4 groups with 3 people in each group, we multiply the individual selections:
Total number of ways = C(12, 3) * C(9, 3) * C(6, 3) * 1
= 220 * 84 * 20 * 1
= 739,200
Therefore, there are 739,200 ways to split 12 people into 4 groups with 3 people in each group.