Question

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 1

Eight celebrities were only on the most intelligent list, and nineteen celebrities were on none of the lists after calculating the unique participants and those not in any category.

Step-by-step explanation:

To find out how many celebrities were only on the most intelligent list and how many were on none of the lists, we can use the principle of inclusion and exclusion. First, let's identify the number of celebrities in each overlap category:

• 6 were on all three lists (powerful, liked, intelligent)
• 19 on the powerful and liked lists
• 13 on the powerful and intelligent lists
• 10 on the liked and intelligent lists

To find how many were only on the most intelligent list, we subtract those who were on multiple lists from the total on the intelligent list:

• Intelligent only = Total on intelligent list - (Powerful & Intelligent) - (Liked & Intelligent) + (All three, because they were subtracted twice)
• Intelligent only = 25 - 13 - 10 + 6
• Intelligent only = 8

Now, let's determine how many were on none of the lists. We add up all the unique members and then subtract from the total number of celebrities:

• Unique members = (Powerful only) + (Liked only) + (Intelligent only) + (Powerful & Liked - All three) + (Powerful & Intelligent - All three) + (Liked & Intelligent - All three) + (All three)
• Powerful only = 41 - 19 - 13 + 6 = 15
• Liked only = 35 - 19 - 10 + 6 = 12
• Intelligent only = 8 (calculated above)
• None = 71 - (15 + 12 + 8 + (19 - 6) + (13 - 6) + (10 - 6) + 6)
• None = 71 - 52
• None = 19

Answer 2

The celebrities on most intelligent list are 29 and 4 celebrities were on none of the list

Step-by-step 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