Question

A new flag is to be designed with six vertical strips using some or all of the colors yellow, green, blue and red. Then number of ways this can be done such that no two adjacent strips have the same color is
$A) 12 \times 81$
$B) 16 \times 192$
$C) 20 \times 125$
$D) 24 \times 216$

Answer 1

Hint: In this question, given that a new flag is to be designed with six vertical strips using some or all of the colours yellow, green, blue and red. First fix a color and find the number of ways this can be done such that no two adjacent strips have the same colour. Finally we get the required answer.

Complete step-by-step solution:
It is given that a new flag is to be designed with six vertical strips using some or all of the colours yellow, green, blue and red.
Here we have to find the number of ways that can be done such that no two adjacent strips have the same color
So now let us take the first step strip, the strip can be coloured in $4$ ways.
Now for the second strip, now if we take anyone colour that is green then colours will be coloured in strips in $3$ ways.
Similarly, for the third, four, five and six that is $3$, $3$, $3$, $3$ ways.
So, now let us find the number of ways this can be done such that no two adjacent strips have the same colour $ = 4 \times 3 \times 3 \times 3 \times 3 \times 3$
Now we need to multiply the expression,
Hence, the number of ways this can be such that no two adjacent strips have the same colour $ = 12 \times 81$
Therefore the number of ways this can be done such that no two adjacent strips have the same colour is $12 \times 81$.

Therefore the correct option is $(A)$.

Note: Probability means possibility. It is a branch of mathematics that deals with the occurrence of random events. So here we have fixed a green colour and solved the problem.
Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur that is how likely they are happening.