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# Gordon buys 5 dolls for his 5 nieces. The gifts include two

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Gordon buys 5 dolls for his 5 nieces. The gifts include two [#permalink]  14 Jun 2010, 05:39
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Hello, a question in Manhattan GMAT guide.

Gordon buys 5 dolls for his 5 nieces. The gifts include two identical S beach dolls, one E, one G, one T doll. If the youngest niece doesn't want the G doll, in how many different ways can he give the gifts?

My ans: Total no. of ways to give gifts = 5P5 /2! = 5!/2! = 60.
How do I account for the youngest niece's condition?
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Re: MGMAT permutations-Combination [#permalink]  14 Jun 2010, 11:10
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rpm wrote:
Hello, a question in Manhattan GMAT guide.
Gordon buys 5 dolls for his 5 nieces. The gifts include two identical S beach dolls, one E, one G, one T doll. If the youngest niece doesn't want the G doll, in how many different ways can he give the gifts?
My ans: Total no. of ways to give gifts = 5P5 /2! = 5!/2! = 60.
How do I account for the youngest niece's condition?

Total # of ways to distribute SSEGT among 5 sisters (without restriction) is \frac{5!}{2!}=60;
The # of ways when the youngest niece gets G is: \frac{4!}{2!}=12 (give G to youngest and then distribute SSET among 4 sisters).

So, # of ways when youngest niece doesn't get G is: 60-12=48.

Hope it's clear.
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Re: MGMAT permutations-Combination [#permalink]  14 Jun 2010, 13:40
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Another way:

Youngest niece needs to choose between S,S, E & T.
case 1) chooses S ==> 4! ways to distribute rest of toys
case 2) doesnt choose S ==> 2 ways * (distribute SSXX to 4 children) = 2 * 4!/2! = 24

Total = 24+24 = 48...
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getting of a doll [#permalink]  16 Feb 2011, 13:21
Pls chk the image

as per my solution

youngest gal can take doll from 5dolls except one.other gal can also take from rest 4 nd so on

4*4*3*2*1=96

But OA is 48
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Re: getting of a doll [#permalink]  16 Feb 2011, 13:43
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Gordon buys 5 dolls for his 5 nieces. The gifts include 2 identical Sun-and-Fun beach dolls, one Elegant Eddie dress-up doll, one G.I. Josie army doll, and one Tulip Troll doll. If the youngest niece doesn't want the G.I. Josie doll, in how many different ways can he give the gifts?

5 nieces: 1 - 2 - 3 - 4- 5
5 dolls: - S - S - E - G- T

1 doesn't want G.

Now if she gets E then the other four dolls (SSGT) can be assigned in 4!/2! ways (permutation of 4 letters out of which 2 S's are identical), the same if she gets T, and if gets S then the other four dolls (SEGT) can be assigned in 4! ways: 4!/2!+4!/2!+4!=48.

Or total was to assign SSEGT to 5 nieces is 5!/2! and ways to assign G to 1 is 4!/2! (the same as E to 1), so desired=total-restriction=5!/2!-4!/2!=48.
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Re: MGMAT permutations-Combination [#permalink]  18 Feb 2011, 02:00
subhashghosh wrote:
Hi Bunuel

I calculated as :

# of ways a doll can be given to the youngest niece = 3 (as 2 are identical in 4)

Then the remaning # of dolls can be distributed in 4!/2! ways.

So total # of ways = 4!/2! * 3 = 36.

Could you please tell me where I'm wrong, i.e, why considering "3" is wrong ?

Regards,
Subhash

If she gets E or T then yes ways to distribute other 4 dolls will be 4!/2!, but if gets she gets S then the other four dolls (SEGT) can be distributed in 4! ways not 4!/2! as all 4 dolls in this case are distinct. So the answer is 2*4!/2!+4!=48 not 3*4!/2!=36.

Hope it's clear.
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Re: MGMAT permutations-Combination [#permalink]  19 Feb 2011, 01:13
Yeah, got it now. Thanks a lot..
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Re: MGMAT permutations-Combination [#permalink]  19 Feb 2011, 12:06
...and ways to assign G to 1 is 4!/2! (the same as E to 1)...

Eh, i understand why its 4!/2! when its 4 dolls to 2 girls when 2 is identical.

buy why its the same if u give her G? can u please explain?

thanks.
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Re: MGMAT permutations-Combination [#permalink]  19 Feb 2011, 12:16
144144 wrote:
...and ways to assign G to 1 is 4!/2! (the same as E to 1)...

Eh, i understand why its 4!/2! when its 4 dolls to 2 girls when 2 is identical.

buy why its the same if u give her G? can u please explain?

thanks.

When you give one doll to the youngest niece you you are left with 4 dolls to assign to 4 sisters. If you give the youngest niece E, G or T then 4 dolls left will have 2 identical S's and # of ways to distribute will be 4!/2! and if you give the youngest niece S then all 4 dolls left will be distinct so # of ways to distribute them will be 4!.

So what's the difference whether you give the youngest niece E or G? In both cases you distribute 4 out which 2 are identical.
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Re: MGMAT permutations-Combination [#permalink]  19 Feb 2011, 13:58
After calculating 5!/2!, you can use the "reduce the pool technique" to find the number of ways in which G doll would be given to the youngest niece.
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Re: getting of a doll [#permalink]  08 Sep 2011, 23:06
GMATD11 wrote:
Pls chk the image

as per my solution

youngest gal can take doll from 5dolls except one.other gal can also take from rest 4 nd so on

4*4*3*2*1=96

But OA is 48

I got 96 too and same way i dont understand when do we have to do 96/2!? Okay, so dolls sun and fun is similar so what? I dont get the concept of dividing by 2?? Please explain ?
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5 Dolls for 5 nieces [#permalink]  30 Jan 2012, 21:50
Gordon buys 5 dolls for his 5 nieces. The gifts include 2 identical "S" dolls, one "E" doll, one "J" doll and one "T" doll. If the youngest niece does not want the "J" doll, in how many different ways can he give the gifts?

Response:

Strategy:
1) Calculate TOTAL number of ways the 5 dolls - S, S, E, J, T - can be assigned to 5 people.
2) Keeping the fifth doll constant ("J"), Calculate TOTAL number of ways that the 4 dolls - S, S, E, T - can be assigned to 4 people.
3) Subtract (2) from (1)

Calculations:
1) 5!/2! = 60
2) 4!/2! = 12
3) 60-12 = 48

Does this seem right? Any other way that you'd approach this?

Book's Response (scroll below)...
.
.
.
.
.
.
48
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Re: 5 Dolls for 5 nieces [#permalink]  30 Jan 2012, 22:00
looks right to me. find total possibility 5!/2! = 60 subtract the chances of of the youngest getting the doll she doesnt want 4!/2! and you get 48
so yea looks right to me. good job.

That should be the only way you approach the problem
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Re: 5 Dolls for 5 nieces [#permalink]  31 Jan 2012, 01:10
Merging similar topics.

fxsunny wrote:
Gordon buys 5 dolls for his 5 nieces. The gifts include 2 identical "S" dolls, one "E" doll, one "J" doll and one "T" doll. If the youngest niece does not want the "J" doll, in how many different ways can he give the gifts?

Response:

Strategy:
1) Calculate TOTAL number of ways the 5 dolls - S, S, E, J, T - can be assigned to 5 people.
2) Keeping the fifth doll constant ("J"), Calculate TOTAL number of ways that the 4 dolls - S, S, E, T - can be assigned to 4 people.
3) Subtract (2) from (1)

Calculations:
1) 5!/2! = 60
2) 4!/2! = 12
3) 60-12 = 48

Does this seem right? Any other way that you'd approach this?

Book's Response (scroll below)...
.
.
.
.
.
.
48

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Re: 5 Dolls for 5 nieces [#permalink]  08 Feb 2012, 11:12
hi bunnuel,

You are explaining the same method again and again. Can u please explain the slot method for this question . From that i too get the answer as 96 ----4*4*3*2*1
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Re: 5 Dolls for 5 nieces [#permalink]  08 Feb 2012, 11:26
vaibhav123 wrote:
hi bunnuel,

You are explaining the same method again and again. Can u please explain the slot method for this question . From that i too get the answer as 96 ----4*4*3*2*1

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Re: 5 Dolls for 5 nieces [#permalink]  08 Feb 2012, 19:51
Bunuel wrote:
vaibhav123 wrote:
hi bunnuel,

You are explaining the same method again and again. Can u please explain the slot method for this question . From that i too get the answer as 96 ----4*4*3*2*1

for the niece who does not want a particular type can be assigned a doll in 4 ways .since 1 doll is assigned out of 5 .Now 4 reminingn can be assigned to other 4 and then 3 to other 3 , then 2 to other 2 and then 1

4*4*3*2*1

I dont get the logic behind when to divide by 2 or not
since if 5 same rings are to be distributed in five fingers, we use the slot method llike this 5*4*3*2*1 and we dont divide it by 5(in this case also 5 rings are identical), so in the above question why we have to divide it by 2.
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Re: 5 Dolls for 5 nieces [#permalink]  09 Feb 2012, 01:43
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vaibhav123 wrote:
Bunuel wrote:
vaibhav123 wrote:
hi bunnuel,

You are explaining the same method again and again. Can u please explain the slot method for this question . From that i too get the answer as 96 ----4*4*3*2*1

for the niece who does not want a particular type can be assigned a doll in 4 ways .since 1 doll is assigned out of 5 .Now 4 reminingn can be assigned to other 4 and then 3 to other 3 , then 2 to other 2 and then 1

4*4*3*2*1

I dont get the logic behind when to divide by 2 or not
since if 5 same rings are to be distributed in five fingers, we use the slot method llike this 5*4*3*2*1 and we dont divide it by 5(in this case also 5 rings are identical), so in the above question why we have to divide it by 2.

Note here that the youngest niece can be assigned a doll in only 3 ways: One of the S dolls (they are both identical so it doesn't matter which one she gets) or E doll or T doll
How you would assign the rest of the dolls would depend on which doll the youngest one got. If she got an S doll, you can assign a doll to the next niece in 4 ways: S or E or G or T. If she got, say, the E doll, you assign a doll to the next niece in 3 ways: S or G or T.
This complicates this method.

Instead, try and assign nieces to the dolls since all nieces are distinct.
G doll can be assigned a niece in 4 ways (the youngest doesn't want her)
E doll can be assigned a niece in 4 ways again (the remaining 4 after one niece has been assigned to G doll)
T doll can be assigned a niece in 3 ways (remaining 3 nieces)
Now we have 2 identical dolls and 2 nieces. How will you assign them? You will give the nieces 1 doll each. There is no other way. Both dolls are same so it doesn't matter who gets which one.

Total number of allocations = 4*4*3 = 48

(or use one of the other great methods discussed above)
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Re: Gordon buys 5 dolls for his 5 nieces. The gifts include two [#permalink]  09 Feb 2012, 02:05
Total number of ways 5C2 = 60 (as on gift is the same)
One does not want to have one specific gift item (Number of ways the niece can get this item) = 4C2 = 12

Total ways to provide gifts = 48 ways
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Re: Gordon buys 5 dolls for his 5 nieces. The gifts include two [#permalink]  20 Feb 2012, 02:41
Hi Bunel ,
Total # of ways to distribute SSEGT among 5 sisters (without restriction) is !5/!2 =60 ;I am trying to understand how did you came to this !5/!2 ?

Is it a permutation of picking 5 out of 5 where 2 are same - 5P2/!2 ?
If this is correct so can it be like if there were 3 sisters instead of 5 , with all other condition intact ,the solution would have been -

Total # of ways to distribute SSEGT among 3 sisters (without restriction) is 5P3/!2 = 15;
The # of ways when the youngest niece gets G is: 4P2/!2 = 6 (give G to youngest and then distribute SSET among 2 sisters).

So, # of ways when youngest niece doesn't get G is:15-6 = 9 .

Please explain for better understanding . Thanks.
Re: Gordon buys 5 dolls for his 5 nieces. The gifts include two   [#permalink] 20 Feb 2012, 02:41
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