# Set Theory, Calendars, Clocks and Binomial Theorem

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## Sets and Unions

In Grand Oberoi hotel, 1160 guests are present currently. The hotel provides the following extra facilities: Gym, Swimming, Fun park, Food. During a regular survey the management team of Oberoi noticed something quite extraordinary about the extra facilities provided by them. They noticed that for every person who uses ‘F’ no. of facilities, there are exactly 3 persons who uses at least (F-1) no. of facilities, F= 2,3,4. They also found that the no. of persons who used no extra facilities is twice the no of person that used all the 4 facilities. Help the management team to find out how many persons used exactly 3 facilities.
1. 40
2. 60
3. 80
4. 100

Choice C. 80

## Detailed Solution

Since, for every person who uses ‘F’ no of facilities, there are exactly 3 person who uses atleast (F-1) no. of facilities. If we take the no. of persons who uses all the four facilities to be ‘x’ then the no. of person who uses atleast 3 facilities will be 3x and so on.
The no. of persons who uses exactly 3 facilities = 3x – x = 2x
Thus, the no. of persons who opt for various facilities can be summarized as follows:

We also know that no of person who uses no facilities = twice of those who uses all the 4 facilities = 2x.
So according the above deductions we can clearly see that the number of person the hotel would be
x + 2x + 6x + 18x + 2x = 29 x = 1160
x = 40
Hence the number of person who uses exactly three science = 2x = 80

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