verycoolguy33 wrote:

Hi Bunuel - Thanks for your reply , but this does not answer my qs. Let me re-phrase my qs .

In the original doll-nieces problem , if there were 3 nieces instead of 5 , so rephrasing the problem -

Q: Gordon buys 5 dolls for his 3 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?

my Ans :

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 .

Thanks,

No, this is not correct. You need to be extremely careful in P&C. One tiny change can change the whole question.

If 5 dolls (S, S, E, G , T) have to be distributed among 3 nieces, how can you distribute them? It depends on which 3 dolls you are distributing. Say you pick S, E, T, there are 3! ways but if you pick S, E, G, you have fewer ways because G cannot go to the youngest niece. So you need to take cases.

Case 1: All dolls distinct. G not included.

Pick 3 of the 4 distinct dolls such that G is not included. You can do it in 1 way: S, E, T

Distribute these 3 among the 3 sisters in 3! ways

Case 2: All dolls distinct. G included.

Pick G and any two of the remaining 3 dolls in 3C2 i.e. 3 ways.

G can be distributed in 2 ways and the rest of the 2 dolls in 2! ways

Case 3: 2 dolls identical and another (not G)

The two identical dolls must be S, S. The unique doll can be chosen in 2 ways.

The unique doll can be distributed in 3 ways and (S, S) will be given to the other 2 sisters in only 1 way since both the dolls are identical.

Case 4: 2 dolls identical and G

The two identical dolls must be S, S.

G can be distributed in 2 ways (to any of the 2 elder nieces)

Then (S, S) can be distributed in only one way.

Total number of ways = 3! + 3*2*2! + 2*3*1 + 2*1 = 26

or do it the reverse way.

Calculate total number of ways of distributing the dolls among 3 nieces and subtract the number of ways in which youngest niece gets G

Total number of ways:

Case 1: All dolls distinct

Select 3 of the 4 distinct dolls in 4C3 ways = 4.

Distribute them among 3 sisters in 3! ways

Case 2: Two dolls identical, one unique

Select S, S and a unique doll in 3 ways.

Give the unique doll in 3 ways (to any of the 3 sisters) and then distribute the identical dolls in 1 way

Total number of ways = 4*3! + 3*3 = 33

Number of ways in which the youngest gets G:

Give doll G to youngest in 1 way.

Now there are 4 dolls {S, S, E, T} and 2 nieces

Case 1: Give distinct dolls

Select 2 out of 3 distinct dolls in 3C2 = 3 ways

Distribute the 2 dolls to the 2 nieces in 2! ways

Case 2: Give identical dolls

Give S, S to the two nieces in 1 way.

Total number of ways in which youngest gets G = 3*2! + 1 = 7

Number of ways in which the youngest doesn't get G = 33 - 7 = 26

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Karishma

Veritas Prep GMAT Instructor

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