Question

How many ways are there to place 3 indistinguishable red chips, 3 indistinguishable blue chips, and 3 indistinguishable green chips in the squares of a 3 x 3 grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
A. 12,
B. 18,
C. 24,
D. 30 ,
E. 36

Answer 1

Answer:

E. 36

Step-by-step explanation:

Let:

$C_1 =$ color 1

$C_2 =$ color 2

$C_3 =$ color 3

Case I

C₂      C₃       C₂

C₃      C₁        C₃

C₁       C₂       C₁

Since C₁ has 3 choices, then C₂ has 2 choices and C₁ has 1 choice

number of combinations = 3 × 2 × 1 = 6

Case II

C₂      C₃       C₁

C₃      C₁        C₂

C₂       C₃       C₁

Since C₁ has 3 choices, then C₂ has 2 choices and C₁ has 1 choice

number of combinations = 3 × 2 × 1 = 6

Case III

C₁      C₃       C₂

C₂      C₁        C₃

C₃       C₂       C₁

Since C₁ has 3 choices, then C₂ has 2 choices and C₁ has 1 choice

number of combinations = 3 × 2 × 1 = 6

Case IV

C₂     C₃         C₁

C₃      C₁        C₂

C₁       C₂       C₃

Since C₁ has 3 choices, then C₂ has 2 choices and C₁ has 1 choice

number of combinations = 3 × 2 × 1 = 6

Case V

C₁      C₂         C₁

C₃      C₁        C₃

C₂      C₃       C₂

Since C₁ has 3 choices, then C₂ has 2 choices and C₁ has 1 choice

number of combinations = 3 × 2 × 1 = 6

Case VI

C₁      C₂         C₃

C₃      C₁        C₂

C₁       C₂       C₃

Since C₁ has 3 choices, then C₂ has 2 choices and C₁ has 1 choice

number of combinations = 3 × 2 × 1 = 6

Therefore, the total combinations = 6 + 6 + 6 + 6 + 6 + 6

= 36