Permutations and Combinations

Question

In a row of 6 counters, 3 are red, 2 are blue and 1 is white. 

 

Find the numver of arrangements in which no red counter is next to another red counter.

Answer 1

Let the counters be indexed 1 through 6. The arrangements where no red counter is next to another will have red counters with indices: (1,3,5) or (1,4,6) or (1,3,6) or (2,4,6). Let us calculate the number of ways in which the first arrangement i.e. red in (1,3,5) can be made - the first counter can be red in 3 ways, the next is not red in 3 ways, the 3rd counter is red in 2 ways, the 4th is not red in 2 ways and the 5th and 6th are not red in 1 way each respectively. The total number of arrangements corresponding to this pattern (1,3,5) is 3*3*2*2*1*1 = 36. The other two patterns can also be arranged in  the same number of ways (i.e. 36 - please work this out yourself if you are not convinced). Therefore the total number of arrangements where no red counter is next to another is 4*36 = 144. If we are after the number of distinct arrangements where red balls (and blue) are indistinguishable from one another, divide this by (3! * 2!) to get 144/12 = 12