Verified answer

★ Concept :-

Here the concept of Permutations has been used. Firstly we see that we are given two figures with 5 parts in them. This means at most only 5 colours can be arranged in them. Then we can calculate the number of ways in which the figure can be coloured if no colours are repeated. Then we can compare the answer for both figures. Then we can check if the colours are repeated. After seeing that, we can again compare and thus find the answer.

Let's do it !!

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★ Solution :-

Given,

» Number in parts in first figure = 5

» Number of parts in second figure = 5

Firstly let's calculate the number of ways in which in which the diagrams can be coloured if no colours are repeated.

This can be done by seeing that there are 5 parts and 5 colours available. This means number of ways will be given as,

→ Number of ways to colour (if no colour is repeated) = 5!

Let's further calculate for the two figures now.

→ Number of ways to colour first figure (if no colour is repeated) = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways

→ Number of ways to colour second figure (if no colour is repeated) = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways

Now let's calculate when the colours are repeated. Firstly we will see the cases and then we will add the result of each case. These will be same for both cases since both have 5 parts and 5 colours.

• When 1 colour is repeated in two parts ::

Now using this condition,

>> Number of ways = 5!/2! = 5 × 4 × 3 = 60 ways

• When 1 colour is repeated in three parts ::

Now using this condition,

>> Number of ways = 5!/3! = 5 × 4 = 20 ways

• When 1 colour is repeated in four parts ::

Now using this condition,

>> Number of ways = 5!/4! = 5 = 5 ways

• When 1 colour is repeated in five parts ::

Now using this condition,

>> Number of ways = 5!/5! = 1 = 1 ways

Now adding the ways we get,

>> Total number of ways (for both figures when colours are repeated) = 60 + 20 + 5 + 1 = 86 ways

Now let's start comparing .

~ When colours are not repeated :-

»»› Number of ways of colouring the first figure = Number of ways of colouring the second figure = 120

~ When colours are repeated :-

»»› Number of ways of colouring the first figure = Number of ways of colouring the second figure = 86

This means the condition is satisfied. So the answer will be Yes.

Yes the number of ways to colour the diagrams is same.

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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