In how many ways can 12 students be put into groups of 3?
1 way,
4 groups of three
C(12,3) which says 12 choose 3
= 12!/(3!9!) = 220
Good calculators have that function build in
Look for nCr
enter 12
2ndF nCr
3
=
you should get 220
misread the question.
I gave you "how many different groups of 3 can you choose from 12"
But you want 4 different groups of 3's, right?
that would be C(12,3) x C(9,3) x C(6,3) x C(3,3)
= 220 x 84 x 20 x 1
= 369600
To find the number of ways to put 12 students into groups of 3, we can use combinatorics.
The first thing we need to do is calculate the number of ways to choose 3 students out of 12 for the first group. This can be done using the combination formula, which is given by:
nCk = n! / (k!(n-k)!)
In this case, n = 12 (total number of students) and k = 3 (number of students per group). Plugging these values into the formula, we have:
12C3 = 12! / (3!(12-3)!) = 12! / (3!9!)
Next, we need to account for the fact that once the first group has been formed, there will be 9 students remaining. We repeat the same process to calculate the number of ways to choose 3 students out of the remaining 9:
9C3 = 9! / (3!(9-3)!) = 9! / (3!6!)
Lastly, once the second group has been formed, there will be 6 remaining students. Again, we calculate the number of ways to choose 3 students out of the remaining 6:
6C3 = 6! / (3!(6-3)!) = 6! / (3!3!)
To find the total number of ways, we multiply these individual combinations together:
12C3 * 9C3 * 6C3 = (12! / (3!9!)) * (9! / (3!6!)) * (6! / (3!3!))
Now let's simplify this expression to find the number of ways:
= (12 * 11 * 10 * 9!) / (3! * 9!) * (9 * 8 * 7 * 6!) / (3! * 6!) * (6 * 5 * 4 * 3!) / (3! * 3!)
= (12 * 11 * 10) / (3 * 2 * 1) * (9 * 8 * 7) / (3 * 2 * 1) * (6 * 5 * 4) / (3 * 2 * 1)
= 220 * 84 * 20
= 369,600
Therefore, there are 369,600 ways to put 12 students into groups of 3.