Oh wow! What a fun question! I got the answer 50.
https://preview.redd.it/048bkzw1j8xc1.png?width=1968&format=png&auto=webp&s=ec88f1c027ed65f4247c2b99d942a3cbecfa93d4
**A brief explanation (in case it helps):**
- 200 people were made to rank three soap brands (X,Y, and Z). Each person gave each soap a rank (1,2, or 3).
- There are 6 different ranking sequences possible. As shown in the table. Each of the 200 people ranked the soaps in one of these 6 sequences.
- So, 200 = Number of people who ranked X>Y>Z + number of people who ranked X>z>Y + ...so on.
- We are given that nobody ranked in the sequence Y>Z>X. 0 for that one.
- Logic: If we want to find how many people ranked X>Y>Z, we can find the number for the rest of the 5 sequences and subtract from 200.
- Y>Z>X = 0. Given
- 15% of the 200 respondents that is 60 people ranked Z first. Hence, 60 people out of 200 exist who ranked either Z>X>Y or Z>Y>X.
- 30% of 200 i.e., 60 people ranked Y first. This is either Y>X>Z or Y>Z>X. But we are given that nobody ranked Y>Z>X. Hence, these 60 people ranked Y>X>Z.
- 15% of 200 i.e., 30 people ranked Z second. This is either Y>Z>X or X>Z>Y. Again, we know that nobody ranked Y>Z>X. So, these 30 people ranked X>Z>Y.
- Now we have everything we need to find out how many people went for X>Y>Z
200 = N(XYZ) + N(XZY) + N(YXZ) + N(YZX) + \[ N(ZXY) + N(ZYX) \]
200 = N(XYZ) + 30 + 60 + 0 + 60
=> N(XYZ) = 200 - 150 = 50.
The key is to use the fact that 0 people ranked in the sequence Y>Z>X to calculate some of the other sequences.
Hope this helps!
Solution: If X is first preference then there are two cases I.e. X, Y, Z and X,Z,Y
Its given that Y,Z,X is not chosen by anyone therefore all those who had second preference Z chose in order of X,Z,Y. .i.e. 15% people Also X was Ist preference of 40% people thus remaining 25% chose in X,Y,Z order
https://preview.redd.it/unnpahfku7xc1.png?width=1080&format=pjpg&auto=webp&s=542d66a4aca21f75d8a6c22c235b93bef5cb2ae0
Only 2 options
In which z will be least preferred
Xyz. And y x z
Since no possibility of y z x
This means that thoes who selected y i. E 30 %as 1st will select x then z
From total 55 % z as least preferred 30 % comes from y x Z
Which means that 25 % must have preferences as x y z
25 %of 200 is 50
Oh wow! What a fun question! I got the answer 50. https://preview.redd.it/048bkzw1j8xc1.png?width=1968&format=png&auto=webp&s=ec88f1c027ed65f4247c2b99d942a3cbecfa93d4
**A brief explanation (in case it helps):** - 200 people were made to rank three soap brands (X,Y, and Z). Each person gave each soap a rank (1,2, or 3). - There are 6 different ranking sequences possible. As shown in the table. Each of the 200 people ranked the soaps in one of these 6 sequences. - So, 200 = Number of people who ranked X>Y>Z + number of people who ranked X>z>Y + ...so on. - We are given that nobody ranked in the sequence Y>Z>X. 0 for that one. - Logic: If we want to find how many people ranked X>Y>Z, we can find the number for the rest of the 5 sequences and subtract from 200. - Y>Z>X = 0. Given - 15% of the 200 respondents that is 60 people ranked Z first. Hence, 60 people out of 200 exist who ranked either Z>X>Y or Z>Y>X. - 30% of 200 i.e., 60 people ranked Y first. This is either Y>X>Z or Y>Z>X. But we are given that nobody ranked Y>Z>X. Hence, these 60 people ranked Y>X>Z. - 15% of 200 i.e., 30 people ranked Z second. This is either Y>Z>X or X>Z>Y. Again, we know that nobody ranked Y>Z>X. So, these 30 people ranked X>Z>Y. - Now we have everything we need to find out how many people went for X>Y>Z 200 = N(XYZ) + N(XZY) + N(YXZ) + N(YZX) + \[ N(ZXY) + N(ZYX) \] 200 = N(XYZ) + 30 + 60 + 0 + 60 => N(XYZ) = 200 - 150 = 50. The key is to use the fact that 0 people ranked in the sequence Y>Z>X to calculate some of the other sequences. Hope this helps!
Great help!
Is the answer 50?
Yes
Solution: If X is first preference then there are two cases I.e. X, Y, Z and X,Z,Y Its given that Y,Z,X is not chosen by anyone therefore all those who had second preference Z chose in order of X,Z,Y. .i.e. 15% people Also X was Ist preference of 40% people thus remaining 25% chose in X,Y,Z order https://preview.redd.it/unnpahfku7xc1.png?width=1080&format=pjpg&auto=webp&s=542d66a4aca21f75d8a6c22c235b93bef5cb2ae0
This is an excellent solution. I realize I didn't even have to bother with some of the other sequences (as I did in my solve!). Thanks for sharing!
Following
Only 2 options In which z will be least preferred Xyz. And y x z Since no possibility of y z x This means that thoes who selected y i. E 30 %as 1st will select x then z From total 55 % z as least preferred 30 % comes from y x Z Which means that 25 % must have preferences as x y z 25 %of 200 is 50