A survey of magazine subscribers placed 71 celebrities in 3 categories. 41 were on the most powerful list. 35 were on the most liked list. 25 were on the most intelligent list. 6 were on all three lists. 19 were on the most powerful and most liked lists. 13 were on the most powerful and most intelligent lists. 10 were on the most liked and most intelligent lists. How many were only on the most intelligent ​list? How many were on none of the​ lists?

Answer: The celebrities on most intelligent list are 29 and 4 celebrities were on none of the list Explanation: According to the question, Criteria 1:  6 were in all three categories  Criteria 2: 19 were placed into most powerful and most liked Criteria 3: 13 were on most powerful and most intelligent  Criteria 4: 10 were on most liked and most intelligent  According to above data,  1) In most powerful category According to Criteria 1, 2 and 3 Total no of celebrities in powerful category = 6+19+13 = 38 No of celebrities left in this category = 45 – 38 = 7 2) In most liked category According to Criteria 1, 2 and 4 Total no of celebrities in most liked category = 6+19+10 = 35 No of celebrities left in this category = 35 - 35 = 0 3) In most intelligent category According to Criteria 1, 3 and 4 Total no of celebrities in intelligent category = 6+13+10 = 29  Therefore, the celebrities on most intelligent list are 29 and 4 celebrities were on none of the list as it is given that only 25 are in most intelligent list