Asurvey 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?

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

the circle one

f is the midpoint of