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805+ (Hard)|   Algebra|      
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At a certain arena, with five levels, level I has 30 vacant seats, 23 Sep 2018, 07:30
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38% (02:09) correct 62% (02:03) wrong based on 217 sessions

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At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. 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?

A) 115
B) 139
C) 140
D) 148
E) 149
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Helium
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At a certain arena, with five levels, level I has 30 vacant seats, 28 Sep 2018, 01:22
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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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AnupamKT
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At a certain arena, with five levels, level I has 30 vacant seats, 23 Sep 2018, 08:26
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Harshgmat
At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. 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?

A) 115
B) 65
C) 140
D) 148
E) 149
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Sir, Kindly solve this.
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pandeyashwin
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At a certain arena, with five levels, level I has 30 vacant seats, 23 Sep 2018, 08:28
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(30-1) + (45-1) + (40-1) + (25-1) + (10-1) + 3 = 148
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At a certain arena, with five levels, level I has 30 vacant seats, 27 Sep 2018, 18:47
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pandeyashwin
(30-1) + (45-1) + (40-1) + (25-1) + (10-1) + 3 = 148
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Hi, please explain in detail. Am lost.
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At a certain arena, with five levels, level I has 30 vacant seats, 27 Sep 2018, 23:06
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Kezia9
pandeyashwin
(30-1) + (45-1) + (40-1) + (25-1) + (10-1) + 3 = 148


Hi, please explain in detail. Am lost.
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"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
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At a certain arena, with five levels, level I has 30 vacant seats, 28 Sep 2018, 00:34
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Harshgmat
At a certain arena, with five levels, level I has 30 vacant seats, level II has 45 vacant seats, level III has 40 vacant seats, level IV has 25 vacant seats and level V has 10 vacant seats. 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?

A) 115
B) 65
C) 140
D) 148
E) 149
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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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At a certain arena, with five levels, level I has 30 vacant seats, 16 Jan 2025, 02:34
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