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boyhappy_5
Nov 06, 2020

Help me please

in Math Problems

Sorry my ability of maths is not so good, could anyone could help me to solve this problem, thanks


There are 3 (k=3) countries teams

(eg, china, japan, korea)


Each team has 5 members.


These ppl need to line up in a row.

But each person should have at least one team member beside him.


How many number of ways can they arrange??


How about the general case when k=n countries?


3 comments
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Ian Fowler
Nov 11, 2020  ·  Edited: Nov 11, 2020

Spent long time on this. Is this from a university course? And if so, what year? I'm curious. This a reasonably difficult problem.


We have:

AAAAA

BBBBB

CCCCC

To obtain at least one member beside another team member they must in:

1) separate pairs and triplets, otherwise at least one will get left out in the cold

2) 5 in a row.


So now we have:

(AAA) (AA)

(BBB)(BB)

(CCC)(CC)


So we now have 6 groups to be arranged

1) 6 choose 2 to place the A's

this also counts the 5 in a row

2) 4 choose 2 to place the B's

3) The C's fall in line. i.e. 2 choose 2

4) 3! to flip the A,B,C's around

this also allows 5 B's in a row and 5C's in a row.

5) Now arrange each of the 5 A's in 5! ways - same for the B's and C's


Finally: (6 choose 2) * (4 choose 2)*(2 choose 2) *(3!) * (5!)^3


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boyhappy_5
Nov 11, 2020

First of all, thank for your message.


But I think there is a little wrong in your step 5.


Eg, if AACCC, then ACACC is one of the case among the 5!.

But the second A has no team member

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Ian Fowler
Nov 12, 2020  ·  Edited: Nov 12, 2020

@boyhappy_5 Hmmm, ... let me think about that. My intention is only to permute a group of 5 like elements which may or may not be 5 in a row - not a set set of 5 mixed elements.

_ A1 A2_ _ _ _ _ A3 A4 A5_ _ _ _


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3 comments