Hey bllnsr.
It's a lot easier to show us what you have tried with these kinds of problems because if the intuition is wrong, then an attempt will show how it is wrong.
Can you show us your attempts?
How to solve these two:
http://i49.tinypic.com/72a8ll.png
And is this question of combination?
http://i50.tinypic.com/ehdb9j.png
Q2: it is written that china mugs are all separated from each other so I placed 6 plastic mugs and 3 china mugs alternately
6!*3!=4320 ways
Q3: This one confuses me and my answer to this question looks way off to me
12!*3!
and the last one I think is a question of combination :
7C3*5C3=350 ways
Hello, bllnsr!
I can't see Q2, but I'll guess what the problem says.
Q2: it is written that china mugs are all separated from each other
so I placed 6 plastic mugs and 3 china mugs alternately: 6!*3!=4320 ways
There are 6 different plastic mugs and 3 different china mugs to be placed in a row
. . where no two china mugs are adjacent.
Place the 6 plastic mugs in row, inserting spaces before, after and between them.
. . . .
There are arrangements of the 6 plastic mugs.
Choose 3 of the 7 spaces and place the 3 china mugs.
. . . There are ways.
Therefore, there are: . possible arrangements.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Suppose the 6 plastic mugs are identical and the 3 china mugs are, too.
Place the 6 plastic mugs in a row with the spaces.
Since they are identical, their order does not matter.
Now select 3 of the 7 spaces and place the 3 china mugs.
. . . There are ways.
Therefore, there are: . possible arrangements.