At a certain arena, with five levels, level I has 30 vacant seats,

You may have gotten tripped up by the wording on this one. The question is not asking for the minimum possible number of seats filled that could completely fill 3 levels, it's asking for the minimum possible number of seats that guarantees that 3 levels will be completely filled. So, first figure out the maximum number of seats that can be filled without completely filling any of the five levels. That means filling all but one vacant seat in each level, that is, 29 in level I, 44 in level II, 39 in level III, 24 in level IV, and 9 in level V, for a total of 145 vacant seats filled. Since there is only one vacant seat in each section now, filling in one more seat will necessarily completely fill a level. Filling in a second additional seat will complete a second level, and filling in a third will complete a third level. So filling in 145 + 3 = 148 seats will ensure that 3 levels are completely filled.
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Sir, Kindly solve this.

(30-1) + (45-1) + (40-1) + (25-1) + (10-1) + 3 = 148

Hi, please explain in detail. Am lost.

"If vacant seats are filled randomly, what is the minimum number of seats that must be filled to ensure that 3 levels are completely filled"

First thing that comes to mind is simply select level 1 , 4, 5 but the seats are filled randomly so you cannot say with 100% certainty that it'll be the case.

So what we do is reduce the seats in all the levels by 1.
So Level 1 = 30-1 = 29 seats ; level 2 = 44 ; level 3 = 39 ; level 4 = 24 ; level 5 = 9.

Now all the levels are on the verge of being filled. Now when the next 3 seats are selected , it'll ensure that 3 levels are completely filled.
(29 + 44 + 39 + 24 + 9 ) + 3 = 148

The seats are to be filled randomly. We need 3 levels to be completely filled. We are looking for the worst case scenario - when will we be sure that 3 levels are completely filled?

Think of the worst case - Each of the 5 levels are filled to (Capacity - 1). Then the last 3 people take seats in 3 levels to finally have 3 levels completely full.
If another 2 people come, all seats would be occupied.

So we are looking for (total seats - 2) = (30 + 45 + 40 + 25 + 10 - 2) = 148

Answer (D)

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