A child is putting white clothes on two dolls. There are three shirts with different shapes, and three pants with different shapes. The shirts and pants that one doll wore should not be put on to the other doll. After putting clothes to two dolls, she colors the shirts and pants with either gray or blue. The colors of the shirts and pants on each doll should be different. How many different possible outcomes are possible?
So the reason I am posting this question is to check my answer as the textbook I am using does not have any answer sheet I can refer to while doing the exercise problems.
This was how I approached the problem:
elements in Sample space = shirts + pants = 3 + 3 = 6.
I interpreted the part of the question where it said:
$$ "The\ shirts\ and\ pants\ that\ one\ doll\ wore\ should\ not\ be\ put\ on \to\ the\ other\ doll" $$
As the set of all possible outcomes for dressing both dolls being mutually exclusive and exhaustive.
Therefore, I took this part to be a simple counting problem where I just square the number of elements(6) to get sample space S:
6*6 = 36.
Now for the next part of the question, where the colors of the shirts and pants are either grey or blue and the colors of both the shirts and pants are different for each doll is the same as the first part of the question except instead of 6, it is 2, therefore,
2*2 = 4.
So for my final answer I multiplied 36 and 4 = 144. Therefore, I said there were 144 possible ways to dress the dolls.
Am I correct? if not, would you be able to explain why? Thank you in advance!
I will give you that by luck somehow You got the right answer but the method you did was wrong.
The sample space is 9 first . Wearing pants and shirt are independent events so have 3*3 ways to be done $(S_1P_1\ ,\ S_1 P_2 ,\ S_1P_3\ ,\ S_2P_1\ ,\ S_2P_2\ ,\ S_2P_3\ ,\ S_3P_1\ ,\ S_3P_2\ ,\ S_3P_3 ).$
Once you dress up first doll the second doll has only 2 options for pant and 2 options for shirt making the sample space 2*2 for 2nd doll. A total of 9*4=36 ways.
Now for the colouring doll 1 has 2*2 ways $(S_bP_b\ ,\ S_bP_g\ ,\ S_gP_b\ ,\ S_gP_g)$. Now once 1st doll has been coloured the sample for 2nd doll 1*1(shirt is left with 1 option and pant is left with 1 option). Here a total of 4 ways.
Answer is 36*4=144.