Around the World in 80 Questions (Day 3): Out of 100 members of the

Ans. (B) 1

93 have a mansion; (M)
90 have a car collection; (C)
81 have a private jet; (J)
75 have a private island; (I)
62 have an art collection. (A)

For finding the least number of the billionaires who have all five items, first take only two items.

Mansion (M) & Car collection (C) -> Both (MC) = 93+90-100 = 83
Now, take MC & J ->MCJ = 83+81-100=64
MCJ & I -> MCJI = 64+75-100=39
MCJI & A -> MCJIA = 39+62-100=1

This is your answer.

93 have a mansion- 7 don't have
90 have a car collection-10 don't have
81 have a private jet-19 don't have
75 have a private island-25 don't have
62 have an art collection-38 don't have
maximum number of people who don't have one time is =7+10+19 +25+38=99
hence the minimum number of people who posses all=100-99=1

In Out Total Out
93 have a mansion 1 7 99
90 have a car collection 8 10 92
81 have a private jet 18 19 82
75 have a private island 37 25 63
62 have an art collection. 62 38 38

Ans B

93 have a mansion -- M
90 have a car collection --C
81 have a private jet -- J
75 have a private island --P
62 have an art collection --A

What is the least number of the billionaires who have all five items ?

Total sample set = n = 100
93 have mansion and 90 have car
the maximum number of people with both will be the least value between them i.e 90
the minimum number of people with both will be 93 - (100- C) = 93 - 10 = 83

Now minimum number of people with M, C and J = 83 (people with both M and C) - (100 - J) = 83 - (100 - 81) = 64

Now minimum number of people with M, C, J and P = 64 - (100 - P) = 64 - (100 - 75) = 39

Now minimum number of people with M, C, J, P and A = 39 - (100 - A) = 39 - (100 - 62) = 1
Least possible value in this sample set with all 5 is 1

Hence answer is B. 1

The highest number of people with all five items is easier to think about. It can only be 62 because any higher number such as 65 is not possible because only 62 people have art collection.
To minimise this number, we can move the groups as apart as possible so that all of them have least intersection.
Let us take the first group- mansion. There are 7 people who don't own a mansion. If these 7 people were among the 62 people, they can never be part of the group with all 5 assets because they don't have a mansion. Now, the possible intersection count reduced to 62-7 = 55.
Similar logic for the other groups gives us
55-(100-90)-(100-81)-(100-75) = 1
B is the right answer.

Total number of members = 100. Let x4 and x5 be number of persons with 4 items and 5 items respectively.

Sum of all sets = 93+90+81+75+62 = 401.

To minimize x5, maximize x4.

Max value of x4 = 100 (as total number of members = 100)

So in this case sum of all persons having items = 4*100 = 400 < 401 so x4 < 100.

Let x4 = 99 and x5 = 1.

In this case sum of all persons having items = 4*99+5*1 = 401. So equation satisfied. So min value of x5 = 1.

IMO B.

Out of 100 members of the billionaires' club "Scrooge McDuck":

93 have a mansion;
90 have a car collection;
81 have a private jet;
75 have a private island;
62 have an art collection.


What is the least number of the billionaires who have all five items ?

A. 0
B. 1
C. 2
D. 7
E. 62


Given that,
93 have a mansion;
90 have a car collection;
81 have a private jet;
75 have a private island;
62 have an art collection.


Total features = 93 + 90 + 81 + 75 + 62 = 401.........(1)

If the billionaires who have all five items is to be minimised, then

the billionaires who have any combination of four items is to be maximised ie, 4 features * 99 billionares = 396 features covered (using suitable features' combinations)

the remaining 100th billionare will be having all five items , thus

Total features = 4*99 + 5*1
= 401 (as per (1))

Hence, least number of the billionaires who have all five items = 1


(B) is the CORRECT answer

Given: Out of 100 members of the billionaires' club "Scrooge McDuck":

93 have a mansion;
90 have a car collection;
81 have a private jet;
75 have a private island;
62 have an art collection.

What is the least number of the billionaires who have all five items ?

A. 0
B. 1
C. 2
D. 7
E. 62

Solution:
We Have 5 types of billionaires i.e. billionaire with one type of luxurious item.

Now to see the least number of billionaires who have all 5 items we want minimum overlap of all five. So we need to spread the 5 around in such a way that all 5 overlap the least.

Out of 100 billionaires, 93 have a or we can say there are 93 mansions. 90 have a car collection means there are 90 car collections.
That means 100 - 93 or 7 billionaires can be allocated a car collection alone and hence remaining 90 - 7 = 83 car collections are allocated to the 93 billionaires who have a mansion.
Hence in the end, we get 83 billionaires with both mansion and car collection, 7 with just a car collection and 10 with just a mansion.

Next 81 have a private jet or there are 81 private jets. We will now allocate the jets to those billionaires who have just 1 item
That means, 7 billionaires now get a car collection and a private jet and 10 billionaires have a mansion and a private jet.
Remaining 81-7-10 = 64 jets will go to those who have a car collection and a mansion.
Hence 64 billionaires now have 3 items while remaining 100 - 64 = 36 billionaires have 2 items each.

Next, 75 billionaires have a private island or say there are 75 private islands. Let us allocate those billionaires first who have 2 items each i.e. 36 billionaires
That leaves 75-36 = 39 private islands left. These islands will have to be allocated to those billionaires who already have 3 items each.
This means we have now 39 billionaires with 4 items each, 61 billionaires with 3 items each.

Lastly 62 billionaires have an art collection meaning there are 62 art collections. We have 61 billionaires that have 3 items each, so allocating 62 art collections to these billionaires leaves us with 62-61 = 1 art collection which will be given to the billionaires who have 4 items each.

Basically we are left with only 1 billionaire who got all 5 items while 99 billionaires got 4 items.

Thus the correct answer is choice B.

% without Mansion = 1 - 93% = 7%
% without car collection = 1- 90% = 10%
% without private jet = 1 - 81% = 19%
% without private island = 1 - 75% = 25%
% without art collection = 1 - 62% = 38%

So % of people not having any of these things = 7 + 10 + 19 + 25 + 38 = 99%

So % of people having it all = 100 - 99 = 1%

Thus, people = 1

Therefore, option B is the correct answer.

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7 - do not have mansion
10 - do not have car collection
19 - do not have private jet
25 - do not have private island
38 - do not have an art collection

Lets say out of 100 , people who do not have art collection is 38, so rest are 62. From 62, 25 do not have private island.
100 - (38+25+19+10+7) = 1


Answer: B

Minimum number of biillionaires that have all 5 items is as follows.

93+90=183
183-100=83
83+81=164
164-100=64
64+75=139
139-100=39
39+62=101
101-100=1

Answer is option B.

7 person doesn't have a mansion,
Least number of people who can have car collection and mansion = 90-7 = 83
Least number of people who can have car collection, mansion, and private jet = 81-17 = 64
Least number of people who can have car collection, mansion, private jet, and private island = 75-36 = 39
Least number of people who can have all the things = 62-61 = 1

IMO B

Answer is B
Assume overlaps between -
93 have a mansion;
90 have a car collection;
81 have a private jet;
75 have a private island;
62 have an art collection
The minimum number of people overlapping between art collection and private island is 75+62-100 = 37
minimum overlap between Art collection,private island and private jet is - 37+81-100 = 118-100 = 18
so on the overlap minimum is 1

93 mansion, 7 without.
90-7 = 83 with mansion, car and 17 others.
81-17 = 64 with mansion, car, private jet and 36 others.
75-36 = 39 with mansion, car, private jet, private island and 61 others.
62-61 = 1 with mansion, car, private jet, private island and art collection and 99 others.
So total = 1.

Answer is B.

The number of people who don’t have the items =7+10+19+25+38=98
Therefore minimum number of billionaires with all items =100-98=2.
The answer would be C.

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the difference between last two 75 have a private island;
62 have an art collection is 7
which is the possibility of at least members having all of the 5 listed items

option D

The Answer is E i.e 62.
Since there are a total of 100 members and 5 items, The minimum possible number of members having all 5 items would be 62.