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# In how many ways can 6 boys be allotted five different rooms such that

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Manager
Joined: 15 Dec 2015
Posts: 120
GMAT 1: 660 Q46 V35
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In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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24 Aug 2017, 06:50
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In how many ways can 6 boys be allotted five different rooms such that none of the rooms are empty and all the 6 boys are accommodated?

A) 6C2 *4!
B) 6C2 *5!
C) 6C5 *5!
D) 6C1 *5!
E) 6C1
[Reveal] Spoiler: OA
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Posts: 5713
Re: In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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24 Aug 2017, 07:22
DH99 wrote:
In how many ways can 6 boys be allotted five different rooms such that none of the rooms are empty and all the 6 boys are accommodated?

A) 6C2 *4!
B) 6C2 *5!
C) 6C5 *5!
D) 6C1 *5!
E) 6C1

HI..

the Q basically means that all rooms will have one boy each except 1 which will have two..
so we find ways to make that pair - 6C2.
and these ocuupants can be accomodated in the 5 rooms in 5! ways..

ans 6C2*5!
B
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Re: In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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24 Aug 2017, 07:54
2
KUDOS
Hi dh99,
Answer would be option B because since there are five rooms. There would be one room allotted to two boys. Now the combination of those two boys would be 6C2. Now assume those two boys as "A" and there would be four boys remaining. Now these four boys and "A" can be allotted in five rooms in 5! Ways. Hence the answer is 6C2*5!

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Manager
Joined: 27 Jan 2016
Posts: 149
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GMAT 1: 700 Q50 V34
Re: In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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24 Aug 2017, 08:08
chetan2u wrote:
DH99 wrote:
In how many ways can 6 boys be allotted five different rooms such that none of the rooms are empty and all the 6 boys are accommodated?

A) 6C2 *4!
B) 6C2 *5!
C) 6C5 *5!
D) 6C1 *5!
E) 6C1

HI..

the Q basically means that all rooms will have one boy each except 1 which will have two..
so we find ways to make that pair - 6C2.
and these ocuupants can be accomodated in the 5 rooms in 5! ways..

ans 6C2*5!
B

Can you pls point out the flaw in my approach.
Selected 5 people in 6c5 ways.
Arranged them in 5! ways
The sixth can occupy any of the 5 rooms. so 5 ways
In total - 6c5*5!*5
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Re: In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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24 Aug 2017, 10:05
chetan2u wrote:
DH99 wrote:
In how many ways can 6 boys be allotted five different rooms such that none of the rooms are empty and all the 6 boys are accommodated?

A) 6C2 *4!
B) 6C2 *5!
C) 6C5 *5!
D) 6C1 *5!
E) 6C1

HI..

the Q basically means that all rooms will have one boy each except 1 which will have two..
so we find ways to make that pair - 6C2.
and these ocuupants can be accomodated in the 5 rooms in 5! ways..

ans 6C2*5!
B

Dear chetan2u , I understand how we get 6C2. However, how did u get the 5! ?
Since we have calculated the first from 5 slots, so we only left with 4 slots --> so the remaining will be 4!.

My asnwer : 6C2 * 4! (A).

What's wrong with my approach?
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Posts: 34
Location: India
Schools: ISB '19 (A)
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GPA: 3.7
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Re: In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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24 Aug 2017, 10:14
septwibowo wrote:
chetan2u wrote:
DH99 wrote:
In how many ways can 6 boys be allotted five different rooms such that none of the rooms are empty and all the 6 boys are accommodated?

A) 6C2 *4!
B) 6C2 *5!
C) 6C5 *5!
D) 6C1 *5!
E) 6C1

HI..

the Q basically means that all rooms will have one boy each except 1 which will have two..
so we find ways to make that pair - 6C2.
and these ocuupants can be accomodated in the 5 rooms in 5! ways..

ans 6C2*5!
B

Dear chetan2u , I understand how we get 6C2. However, how did u get the 5! ?
Since we have calculated the first from 5 slots, so we only left with 4 slots --> so the remaining will be 4!.

My asnwer : 6C2 * 4! (A).

What's wrong with my approach?

The room with 2 boys can be chosen in 5C1 ways.
So the total combinations become 6C2*4!*5=6C2*5!.
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Joined: 21 Sep 2016
Posts: 28
Re: In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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11 Sep 2017, 18:26
Out of 6 elements we have to pick 1 pair, so there will be 6C2 ways of choosing a pair.

Now, we have 5 elements: 1 pair of boys and the remaining 4 boys. Since there are 5 rooms to be occupied, we have 5! ways of allotting those 5 elements.

Thus: $$6C2 * 5!$$
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In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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11 Sep 2017, 23:06
1
KUDOS
septwibowo wrote:
chetan2u wrote:
HI..

the Q basically means that all rooms will have one boy each except 1 which will have two..
so we find ways to make that pair - 6C2.
and these ocuupants can be accomodated in the 5 rooms in 5! ways..

ans 6C2*5!
B

Dear chetan2u , I understand how we get 6C2. However, how did u get the 5! ?
Since we have calculated the first from 5 slots, so we only left with 4 slots --> so the remaining will be 4!.

My asnwer : 6C2 * 4! (A).

What's wrong with my approach?

Hi septwibowo,

While solving this problem we are able to choose 2 of the 6 members
A,B,C,D,E and F using 6c2(it could be AB/AE or any such combination)
Once they become room mates, we will have 5 groups, 4 individuals and 1 team(call them X)
and 5 rooms to fill them.

We can fill the people in the rooms in 5! ways as there are 5 groups(and 5 rooms)
So the final answer should be 6C2 * 5!(Option B)

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Re: In how many ways can 6 boys be allotted five different rooms such that [#permalink]

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13 Sep 2017, 08:30
srikanth9502 wrote:
chetan2u wrote:
DH99 wrote:
In how many ways can 6 boys be allotted five different rooms such that none of the rooms are empty and all the 6 boys are accommodated?

A) 6C2 *4!
B) 6C2 *5!
C) 6C5 *5!
D) 6C1 *5!
E) 6C1

HI..

the Q basically means that all rooms will have one boy each except 1 which will have two..
so we find ways to make that pair - 6C2.
and these ocuupants can be accomodated in the 5 rooms in 5! ways..

ans 6C2*5!
B

Can you pls point out the flaw in my approach.
Selected 5 people in 6c5 ways.
Arranged them in 5! ways
The sixth can occupy any of the 5 rooms. so 5 ways
In total - 6c5*5!*5

Can someone please explain why this approach is incorrect? I also followed the same approach.
Re: In how many ways can 6 boys be allotted five different rooms such that   [#permalink] 13 Sep 2017, 08:30
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