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

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Manager
Joined: 21 Sep 2008
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At a certain arena, with five levels, level I has 30 vacant [#permalink]

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20 Nov 2008, 21:35
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
any good explanation?
From: Kaplan PS: Statistics and Probability, Part 2 - Challenge Question:4 of 10

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20 Nov 2008, 22:34
D. 148

here's how:

consider it this way, we have only 2 rows having 10, 15 seats respectively and we have to assure that one row is filled up completely.
if we have 15 people, they can seat in 8 + 7 seats
if we have 20 people, they can seat in 8+ 12 seats
if we have 22 people still they can seat in 9+ 13 seats
if we have 23 people still they can seat in 9 + 14 seats
This is the saturation point, addition of one person to this will make either row full.
so if we have 24 ppl, one row must be full

Observe that saturation point is reached when we have
(capacity of row - 1 person) in "each" row.
or (total seats - #of rows) people
&
addition of every one person gaurantees that additional one row is getting full.

here we have 30 45 40 25 10 = 150 total seats.
we have 5 rows => 150 - 5 = 145 is saturating point.
& add one person and it gaurantees that one row is full.
we have to assure that 3 rows are full.

so we have to add 3 people to saturating point

145 + 3 = 148 minimum people will be needed

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Re: Kaplan Probability question   [#permalink] 20 Nov 2008, 22:34
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# At a certain arena, with five levels, level I has 30 vacant

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