Question

A flag is to be coloured in four stripes by using 6 different colours, no two consecutive stripes being of the same colour. This can be done
(a) 1500 ways
(b) 750 ways
(c) ${{6}^{4}}$ ways
(d) None of these

Answer 1

Let us start by filling the stripes one by one. The first stripe has 6 options of colour, while the second has 5 as cannot have the same colour as the previous stripe. Similarly, third and fourth have 5 options each as cannot have the same colour as the previous one, because it is mentioned that no two adjacent stripes have the same colour. As all the strips are to be painted at the same time so multiply the option.

Complete step-by-step answer :
It is given that there is a flag which is to be coloured and consists of 4 stripes. We are asked to fill colours in each of the stripes with given 6 colours such that no two adjacent stripes have the same colour. So, let us draw a representative diagram of the situation.

So, we have 6 options of colour to fill the first stripe. Now we have only 5 options to fill the second stripe as we cannot fill it with the same colour as 1. Similarly, stripe 3 has 5 options, one colour that is filled in stripe 2 cannot be filled in it. Also, stripe 4 will have 5 options, one colour filled in stripe 3 cannot be filled here.
Now, for finding the number of ways of painting all these stripes together, we have to multiply the number of options for each stripe.
$\text{number of ways}=6\times 5\times 5\times 5=750$
Hence, the answer to the above question is option (b).

Note :The first thing you should be very careful is to keep in mind the constraint that no two consecutive stripes have the same colour. If you want you can also find the answer to the above question by finding the number of filling the stripes without any constraint and subtracting those cases in which two adjacent stripes have the same colour.