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In a locality, there are ten houses in a row. On a particular night

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In a locality, there are ten houses in a row. On a particular night  [#permalink]

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New post 24 Jul 2017, 18:56
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In a locality, there are ten houses in a row. On a particular night a thief planned to steal from three houses of the locality. In how many ways can he plan such that no two of them are next to each other?
A. 56
B. 73
C. 80
D. 120
E. None of the above

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In a locality, there are ten houses in a row. On a particular night  [#permalink]

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New post 24 Jul 2017, 19:27
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Let the houses the thief has planned to steal from be A,B,C, and the other houses be X1,X2,...X7.

Since A,B,C cannot be together, the remaining 7 houses can be placed in the following manner: -X1-X2-X3-X4-X5-X6-X7-, where the "-" denotes the place that can be filled with A,B, or C.

Since there are 8 "-"s and 3 items to fill, the no. of ways this can be done is 8c3 = 56. Ans - A.

Another quick way to work through these sort of problems is to deduct the no. of items that cannot be placed together from the total no. of items, then add 1 and do the combinations. In this case, 10-3+1 = 8c3.
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In a locality, there are ten houses in a row. On a particular night  [#permalink]

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New post 24 Jul 2017, 19:45
If there are 10 houses and the thief needs to choose from 3 houses,
the total ways in which he can choose 3 houses are \(10c3 = \frac{10*9*8}{3*2} = 120\) ways
The condition is that he can't steal from 2 houses that are next to each other.

One way of finding that out is finding how many such combinations are there are reducing that number
from the total combinations possible, which is 120.

To find out how many ways are there if he steals from houses that are next to each other.
If the houses are next to each other, there are 9 ways of choosing the first 2 houses,
and 8 ways of choosing the next house, so a total of 9*8(72) such combinations.
However, there will be 8 combinations which will end up getting counted twice because

_ _ _ _ _ _ _ _ _ _
is the same as
_ _ _ _ _ _ _ _ _ _

That makes a total of 64 combinations(72 - 8) where the houses he steals from are next to each other.

Hence, the total possibilities are 120 - 64 = 56(Option A)

--== Message from the GMAT Club Team ==--

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This discussion does not meet community quality standards. It has been retired.


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Re: In a locality, there are ten houses in a row. On a particular night  [#permalink]

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New post 14 Oct 2018, 01:52
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Re: In a locality, there are ten houses in a row. On a particular night   [#permalink] 14 Oct 2018, 01:52
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