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