Question

There is a set of chips of 5 different colours (there are at least 6 chips of every colour). It is necessary to put 6 of these chips in a row (from left to right) in such a way that any two adjacent chips are of different colours, and at least three colours have to be used. How many ways of doing it are there?

Answer 1

The number of ways when all adjacent chips are of different colours: On the first place we can put any colour, on the others - any except of the colour of the previous one, so $k=6*5*5*5*5*5=18750$

But there can be a situation when we use only 2 colours. The number of ways when all adjacent chips are of different colours and onle 2 colours is used is: We choose two different colours to be placed on the first two places, the colours on the other places will be determined then, so $m=6*5*1*1*1*1=30$

So, the total number of ways is $n=k-m=18750-30=18720$