Maths Extension probability question (1 Viewer)

[KikiKai04]

Six people are to be divided into two groups, each with at least one person. Find the probability that:
a. there will be three in each group,
b.there will be two in one group and four in the other,
c. there will be one group of five and an individual

How do i solve this?

 

[CM_Tutor]

Suppose we have the six people in a row, and each is assigned to either group A or B.

Since there must be at least one in each group, I can get "words" of

1 x A and 5 x B, and there are 6 such groups possible

2 x A and 4 x B, and there are 15 such groups possible

3 x A and 3 x B, and there are 20 such groups possible

4 x A and 2 x B, and there are 15 such groups possible, but as the goal is merely separate groups, there is no difference between persons 1, 2, 3, and 4 in group A and persons 5 and 6 in group B than in having persons 1, 2, 3, and 4 in group B and persons 5 and 6 in group A, so these 15 groups exactly duplicate the 15 groups with 2 x A and 4 x B

5 x A and 1 x B, which exactly duplicates the 6 groups with 1 x A and 5 x B

So, there are 6 + 15 + 20 = 41 groups.

Assume that every group is as likely as every other group, then: