Question

A game designer must decide how to color five buildings that are in a row. Using only the colors yellow, green, red, and blue, each building must be painted with exactly one color. Any two neighboring buildings must be different colors, and the first, third, and fifth buildings must be different colors. How many ways are there to paint the five buildings?

Answer 1

84

Explanation:

we can split it into 2 cases: 1 case is when you choose the last building first and the second case is when you choose it last: so it is 4x2x2x3 for the first case and 4x3x3x1 for the last. So it’s 36+48=84 total cases.

Answer 2

The total of ways the buildings can be painted are = 4
`*` 3
`*` 2
`*` 2
`*` 1 = 48 ways.

Explanation:

i) color five buildings that are in a row.

ii) any two neighboring buildings must be different colors

iii) the first third and fifth buildings must be different colors.

iv) the number of ways to paint the fist building are 4.

v) the number of ways to paint the second building are 3.

vi) the number of ways to paint the third building are 2.

vii) the number of ways to paint the fourth building are 2.

viii) the number of ways to paint the last building are 1.

ix) therefore the total of ways the buildings can be painted are

= 4
`*` 3
`*` 2
`*` 2
`*` 1 = 48 ways.