Tie CombinationsDate: 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/ |
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