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In a survey of customer satisfaction, participants are asked to give a score of 1,2,3 or 4 to each of the 6 questions. If participants are instructed not to give the same numerical score to more than 4 questions, how many responses are possible?? I know the answer is 4,020 from the solutions, however I don't understand why. The solution says: $$4*3*{6 \choose 5}=72$$ Hence, the number of possible responses is: $$4,096-4-72=4,020$$ Why and how is this the answer? I don't understand where the $-4$ comes from.

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up vote 4 down vote accepted

If we ignore the restriction on the number of common responses, there are $4^6=4096$ possible responses. Now we need to subtract those that violate the requirement of no more than $4$ common ones. It can be violated if all the responses are the same, which can happen in $4$ ways, or if five are the same and one different. To have five the same, you can select the common response in $4$ ways, the odd response in $3$ ways, and the question number of the odd response in $6 \choose 5$ ways. This gives the book answer as the number of valid responses.

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