Question

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ______

Answer 1

Five persons; A, B, C, D and E.

So, total number of persons = 5

All of them are seared in circular arrangement. Each of them is given a hat of one of the three colours red, blue and green.

Total number of caps = 3 (Red, Blue and Green)

But there are total five person. So, they can distributed as Red, Blue, Blue, Green, Green.

As there are only three coloured caps and five person. So, two caps are of same colour.

Now, number of ways in which caps can be distributed is 3.

→ Red, Blue, Blue, Green, Green.

→ Red, Red, Blue, Green, Green.

→ Red, Red, Blue, Blue, Green.

Now,

Total number of ways = Total number of persons × Total number of caps × Total number of ways in which caps can be distributed

⇒ 5 × 2 × 3 = 30 ways

Answer 2

Given :-

• 5 persons A, B, C, D and E are seated in a circular arrangement...
• 3 hat of colours = Red , blue and Green.

To Find :-

• the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats ?

Answer :-

From Given Data, we can say That:-

→ Maximum Number of Hats we can use of Same color = 2.

Reason :- if we used More Than 2 Hats of same color ,

Than, Atleast 2 hats of same color will be consecutive.

______________________

So,

→ Hats can be used in 3 ways :-

❶ R -- R -- G --- B --- B

❷ R -- R -- G --- G --- B

❸ R -- G -- G --- B --- B

_________________

Now,

Case (1) :-

→ Number of Ways of Selecting 1 Blue Hat is = 5 person and 5 hats = Let A received Blue Hat.

Case (2) :- (This one Is Tricky).

→ Now, we can say That = Either B & D are Filled with Red hats and Rest two (C & E) are filled with Green Hats.

OR,

Either B & D are Filled with Green hats and Rest two (C & E) are filled with Red Hats.

= 2 ways Are Possible.

__________________

So,

☛ Total Number of Possible ways = (3 ways in which Hats used ) * (5 ways in Which Blue hat used ) * 2 ways in which (Red & Green hats used) = 3 * 5 * 2 = 30 ways. (Ans).

Hence, the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is 30.