Tie Combinations

Date: 05/31/99 at 15:02:30
From: Bob Feller
Subject: Combinations

If four people have three identical ties, what is the total number of 
different combinations the four can wear? No repeats; in other words, 
AAB and BAA count as one. 

What is the probability that all four will wear the same tie? 

Thanks.

Date: 05/31/99 at 19:12:05
From: Doctor Anthony
Subject: Re: Combinations

If the ties are denoted by a, b, c, then we require the number of 
different ways of making up 4 letters from the 3 available a, b, c. 
For example  aabc, abbc,  aaaa, abcc are all possibilities.

Partitions of 4 with no more than 3 parts:
-----------------------------------------
  1,1,2  choose the 2 in 3 ways  
  4      choose the colour in 3 ways
  2,2    choose the 2 colours in C(3,2) = 3 ways
  3,1    choose the 3 in 3 ways and the 1 in 2 ways = 6 ways 

I am assuming that it does not count as a different arrangement if we 
have the same colour scheme but with different people wearing the 
colours.

Total number of colour schemes = 3 + 3 + 3 + 6 = 15 (but they are not 
eqi-probable).


Probability that all 4 wear the same colour.
-------------------------------------------
We must be careful to consider equi-probable events.

The first person could choose a colour in 3 ways; then the others must 
choose the same colour.

The total ways that all 4 could choose colours is 3^4 = 81.

So the probability that they will all wear the same colour is 
3/81 = 1/27.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/