★ Concept :-

Here the concept of Permutations have been used. We see that we are given a figure with 5 parts in it. We need to find the possibilities. The confusion arises when people think to use Probability here but that's actually wrong. Here it has been asked 'how many possibilities' so clearly, we need to find the number of possibilities. Firstly here we shall understand the figure. We will look to the different cases and thus find the answer.

Let's do it !!

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★ Solution :-

Given,

» There is 1 figure with 5 parts in it namely A, B, C, D and E.

1.) 5 colours are available ::

We see that we need to colour the whole figure from part by part. We are given with 5 colours. It's not given that the parts are filled by one colour each. This means that every part has possibility to be filled by 5 colours. Then we will get the number of ways as 5 raised to 5.

>> Number of possibilities = 5⁵ = 5 × 5 × 5 × 5 × 5 = 3125 possibilities.

• Total number of possibilities = 3125

2.) The neighbouring region should be in different colour.

The neighbouring region should be in different colour. We see that each part is atmost indirectly related with other parts. Let's the relationship :-

• A has neighbouring parts as B, C, D and E. This means all these parts should have different colour than A.

• B has neighbouring parts as A, C and E. This means all these parts should have different colour than B.

• C has neighbouring parts as A, D and E. This means all these parts should have different colour than C .

• D has neighbouring parts as C, A and E. This means all these parts should have different colour than D.

• E has neighbouring parts as A, B and D. This means all these parts should have different colour than E.

From these information (derived from given figure) we get that all the parts should have different colours. Now we have the permutation to calculate the number of possibilities.

>> Number of possibilities = 5! = 5 × 4 × 3 × 2 × 1 = 120 possibilities

This is the answer.

• Total number of possibilities = 120 .

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★ More to know :-

• 1! = 1

• 2! = 2 × 1

• 3! = 3 × 2 × 1

• 4! = 4 × 3 × 2 × 1

• 5! = 5 × 4 × 3 × 2 × 1

• 6! = 6 × 5 × 4 × 3 × 2 × 1

• 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

• 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

• 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

• 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Verified answer

Answer:

1.Since 5 regions are out there namely A,B,C,D and E.

Therefore possibility of 5 colours are available is 5/5=1

2.Now possibility of neighbouring regions should be different regions=4/5=0.8