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A parking lot has 16 spaces in a row. Twelve cars arrive, each of whic

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A parking lot has 16 spaces in a row. Twelve cars arrive, each of whic  [#permalink]

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New post 20 Mar 2019, 05:59
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A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose their spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?

A) \(\frac{11}{20}\)

(B) \(\frac{4}{7}\)

(C) \(\frac{81}{140}\)

(D) \(\frac{3}{5}\)

(E) \(\frac{17}{28}\)
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Re: A parking lot has 16 spaces in a row. Twelve cars arrive, each of whic  [#permalink]

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New post 20 Mar 2019, 06:21
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Noshad wrote:
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose their spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?

A) \(\frac{11}{20}\)

(B) \(\frac{4}{7}\)

(C) \(\frac{81}{140}\)

(D) \(\frac{3}{5}\)

(E) \(\frac{17}{28}\)



OK... We have 16 spaces and 12 occupy spaces, so the 4 vacant can be in 16C4 ways.
Now, it is easier to find combinations or cases when the 4 vacant are not next to each other.
For this, place 12 SUVs first, so there will be 13 places for these 4 vacant places, so 13C4. => _1_2_3_4_5_6_7_8_9_10_11_12_

So, Probability that the vacant places are available = \(\frac{16C4-13C4}{16C4}=\frac{16*15*14*13-13*12*11*10}{16*15*14*13}=\frac{13*12*10(28-11)}{28}=\frac{17}{28}\)

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A parking lot has 16 spaces in a row. Twelve cars arrive, each of whic  [#permalink]

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New post 22 Mar 2019, 05:47
can you please let me why my logic is wrong :
total number of ways : 16C4
now let's count available ways for 2 adjacant spaces :
let a :2 adjacent places as a unit , b : one space , c : a car
now we have abbccccccccc ,let's arrange them in 15!/(12!*2!)
we divided by 12! because c is repeated , the same for 2! ( b is reapeted )
thus p = {15!/(12!*2!)} / 16C4
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A parking lot has 16 spaces in a row. Twelve cars arrive, each of whic   [#permalink] 22 Mar 2019, 05:47
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