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Five friends go to a city where they can stay in 10 different hotels.

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chetan2u
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Five friends go to a city where they can stay in 10 different hotels. [#permalink] New post   30 Apr 2016, 00:54
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Five friends go to a city where they can stay in 10 different hotels. In how many ways can they stay in these hotels if atleast two of them stay together?
(A) \(5^{10} - 10C5*5!\)
(B)\(5^{10} - 10C5\)
(C)\(5^{10} - 5!\)
(D) \(10^5 - 10C5*5!\)
(E) \(10^5 - 10C5\)

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Re: Five friends go to a city where they can stay in 10 different hotels. [#permalink] New post   30 Apr 2016, 01:21
b

Each has the option of staying in 10 different hotels ,total number of ways=5^10
This also includes combination where each stays in 5 different hotels=10c5
5^10 - 10c5
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Re: Five friends go to a city where they can stay in 10 different hotels. [#permalink] New post   30 Apr 2016, 01:24
how many ways can they stay in these hotels if atleast two of them stay together --> Total - Number of ways so that no person stays together

There are 10 hotels and each person can stay in 10 hotels

Total = 10 * 10 * 10 * 10 * 10 --> 10^5 --> Notice that there is no restriction to avoid a hotel.

Number of ways so that no person stays together with another friend = 10 * 9 * 8 * 7 * 6 = 10P5 = 10C5 * 5!

Number of ways can 5 friends stay in these hotels if atleast two of them stay together = 10^5 - (10C5 * 5!)

Answer: D
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Re: Five friends go to a city where they can stay in 10 different hotels. [#permalink] New post   30 Apr 2016, 01:30
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goforgmat wrote:
b

Each has the option of staying in 10 different hotels ,total number of ways=5^10
This also includes combination where each stays in 5 different hotels=10c5
5^10 - 10c5


Hi
its very important to decide \(5^10 or 10^5\), YES it is confusing..
think of it this way-
You will place friends in different hotels, so 10 hotels so each friend can go into any of 10 hotels, so 5 friends = any of 10 * any of 10 *any of 10 * any of 10 * any of 10 = \(10^5\)
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Re: Five friends go to a city where they can stay in 10 different hotels. [#permalink] New post   27 Jul 2020, 04:52
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Re: Five friends go to a city where they can stay in 10 different hotels. [#permalink]
27 Jul 2020, 04:52
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