Question

What is the number of ways to color objects with 3 colors if every color must be used at least once?

Answer 1

Given : every color must be used at least once

3 colors

To Find :  number of ways to color objects

Solution:

if there are 3 objects  the 3 colors  can be used  

in 3!  = 6 ways  

to color as  every color must be used at least once

if there are n objects then number of ways

=  3ⁿ - 3.2ⁿ   + 3

3ⁿ  is the case where no restriction which color

3.2ⁿ  is with just 2 colors   which include  3.2  = 6 cases with 1 color

3  is with 1 color    

Hence 3ⁿ  - (3.2ⁿ - 6)  - 3  =    3ⁿ - 3.2ⁿ   + 3

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Answer 2

Given :- What is the number of ways to color objects with 3 colors if every color must be used at least once ?

Answer :-

we know that,

• The number of ways to color objects with n colors if every color must be used at least once is n ! .

given that,

• n = 3 .

then,

→ The number of ways to color objects with 3 colors if every color must be used at least once is = 3! = 3 * 2 * 1 = 6 ways (Ans.)

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