The question looks a bit data heavy, but one strike at the right place immediately gives you the answer.

GivenWe are given a set of data with user's preferences of three brands of soaps X, Y & Z. We are told that 200 people responded to the survey with none of them having preference in the order Y,Z,X. We are asked to find the number of respondents who rated the soap in the order of X,Y,Z.

ApproachWe are given total number of responses of people at each rank.

The key to note here is that people can only vote for soaps in a rank for which they have not voted earlier. For example, if a person has voted soap X as rank 1, he can't vote for soap X again in any of the subsequent ranks. As an extension to this, we can say that people who voted for soap x as rank 2 would have voted for either soap Y or soap Z as rank 1.

We will use this concept along with the fact that no one ranked the soaps in the order of Y, Z, X to find out the number of people who ranked the soaps in the order of X, Y, Z.

Working OutWe are given that no one ranked the soaps in the order Y,Z,X. We also know that 60 i.e. 30% of 200 people ranked soap Y as 1. People who ranked soap Y as 1 did not rank soap Z as 2 which implies that all of them ranked soap X as 2. These people then further ranked soap Z as 3. Note here that these people can't vote for soap X or soap Y as rank 3 because they have earlier ranked soap X & soap Y.

We also know the total number of people who ranked soap Z as 3 is 55% of 200 i.e. 110. Soap Z can be ranked 3 by people who voted for soap X or soap Y as rank 1. Out of these we know that 60 people who voted for soap Y as rank 1, voted for soap Z as rank 3. This leaves 110 - 60 = 50 people who voted for soap Z as rank 3. This can only come from people who voted for soap X as rank 1 and then soap Y as rank 2.

Thus, we know that there were 50 people who ranked in the soaps in the order X, Y, Z

The other blocks in the tree structure do not need to be calculated (although they hardly take any time). They have been calculated here just for your reference.

Hope its clear!

Regards

Harsh

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