Question

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

Answer 1

Hint: We are 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 colour and find the number of ways cloured for each strip and after that find, the number of ways this can be done such that no two adjacent strips have the same colour .

Complete step-by-step answer:
Now we are 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.
So now let us take the first strip, the first strip can be coloured in $4$ ways.
Now for the second strip, now if we take anyone colour i.e. green then remaining colours will be coloured in strip in $3$ ways.
Similarly, for the third strip we have again $3$ ways.
Also, same goes for strip four, five and six i.e. in $3$ 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 simplifying above we get,
The number of ways this can be done 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 $12\times 81$.
The correct answer is option (A).

Additional information:
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Note: Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. So here we have fixed a green colour and solved the problem.