  #### 1600 satellites were sent up by a country for several purposes. The purposes are classified as broadcasting (B), communication (C), surveillance (S), and others (O). A satellite can serve multiple purposes; however a satellite serving either B, or C, or S does not serve O. The following facts are known about the satellites: 1. The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2: 1:1. 2. The number of satellites serving all three of B, C, and S is 100. 3. The number of satellites exclusively serving C is the same as the number of satellites exclusively serving S. This number is 30% of the number of satellites exclusively serving B. 4. The number of satellites serving O is the same as the number of satellites serving both C and S but not B. Question: If the number of satellites serving at least two among B, C, and S is 1200, which of the following MUST be FALSE? Option: 1 The number of satellites serving C cannot be uniquely determined   Option: 2 The number of satellites serving B is more than 1000   Option: 3 All 1600 satellites serve B or C or S   Option: 4 The number of satellites serving B exclusively is exactly 250  Given data:

1. Number of satellites serving B, C, and S (though maybe not exclusively) are in the ratio

2. Number of satellites serving all three of B, C, and S is 100

3. Number of satellites exclusively serving C is the same as the number of satellites exclusively serving S (= 30% of the number of satellites exclusively serving B)

4. Number of satellites serving O is the same as the number of satellites serving both C and S but not B

Since there are satellites serving a single or multiple purposes, Venn Diagram is a good method to solve.

For ease of calculation, let the number of satellites exclusively serving B = 10x.

the number of satellites exclusively serving C and S Now, let the number of satellites serving others(O) by y. Lastly, let the number of satellites serving B, C but not S be z.

Since the numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio

number of satellites serving B, S but not C = z. It is given that the total number of satellites =1600 :

The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2:1:1

Solving the equations (substitute this in equation 1): Using the boundary condition for x,

Also, Therefore, number of satellites in each of the category can be determined.

Hence, option C is definitely false.

Therefore, we can say that option C is incorrect.