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# Numbers and Counting question watch

1. I am confused by the following question, the second part of (a) and (b). For (a) I got 560, which I know is correct. When it talks about removing the hats, I got the answer 28, when the solutions say it's 280.

14. (a) 8 people are to be divided into three teams, distinguished by the colour hats they wear.
2 people will wear red hats, 3 green hats and 3 blue hats. How many ways are there of
choosing the teams? What if we remove the hats so that the two teams of size three become
interchangeable?
(b) 8 people are to be divided into four teams, distinguished by hat colour: 2 people will wear
red hats, 2 green hats, 2 blue hats and 2 orange hats. How many ways are there of choosing
the teams? What if we remove the hats?

Anyone help?
2. (Original post by theereflex)
I am confused by the following question, the second part of (a) and (b). For (a) I got 560, which I know is correct. When it talks about removing the hats, I got the answer 28, when the solutions say it's 280.

14. (a) 8 people are to be divided into three teams, distinguished by the colour hats they wear.
2 people will wear red hats, 3 green hats and 3 blue hats. How many ways are there of
choosing the teams? What if we remove the hats so that the two teams of size three become
interchangeable?
(b) 8 people are to be divided into four teams, distinguished by hat colour: 2 people will wear
red hats, 2 green hats, 2 blue hats and 2 orange hats. How many ways are there of choosing
the teams? What if we remove the hats?
The second part of 14(a) means the following scenarios are now the same since there are no hats:

Green hats: A, B, C
Blue hats: X, Y, Z

Green hats: X, Y, Z
Blue hats: A, B, C

Hence there should be half the number of possibilities in the first part of 14(a).

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