Question

In: Advanced Math

Let an denote the number of different ways to color the walls of a five-sided room...

Let a_n denote the number of different ways to color the walls of a five-sided room with n colors if you insist that two walls that meet at a corner must be assigned different colors.

(i) compute a1, a2 and a3 directly

(ii) Find the formula for a_n

Expert Solution

Assume there are 5 walls in the room naming A, B, C, D and E in the order.

(i)

For n=1, there is only 1 color present.

If we paint first wall then, there is no other color left for another wall. So,

number of ways=0

a₁=0

For n=2, there are 2 colors, C1 and C2 are present.

If we paint wall A with color C1 then its surrounding walls, B and E must be of another color C2.

So, C should be of color C1.

Wall D cannot be of color C2 because of wall E nor of color C1 because of wall C.

So,

Number of ways=0

a₂=0

For n=3, there are 3 colors, C1, C2 and C3 are present.

If we paint wall A with color C1 (out of 3 options) then its surrounding walls, B and E must be of another color C2 (out of 2 options each).

So, C could be of color C1 and C3 (2 options).

Wall D cannot be of color C2 because of wall E. If wall C is of C1 color then wall D will be of C3 color and if wall C is of C3 color then wall D will be of C1 color.

Thus, 1 option.

So,

number of ways= (options of coloring wall A) (options of coloring wall B) (options of coloring wall C)(options of coloring wall D) (options of coloring wall E

a₃=(3)(2)(2)(2)(1)

=24

(ii)

For any number n, there are n colors, C1, C2 ,C3… C_n are present.

If we paint wall A with color C1 (out of n options) then its surrounding walls, B and E must be of another color C2 (out of n-1 options each).

So, C could be of color C1 and C3 (n-1 options).

Wall D cannot be of color C2 because of wall E. If wall C is of C1 color then wall D will be of C3 color and if wall C is of C3 color then wall D will be of C1 color.

Thus, n-2 option.

So,

number of ways= (options of coloring wall A) (options of coloring wall B) (options of coloring wall C) (options of coloring wall D) (options of coloring wall E)

a_n=(n)(n-1)(n-1)(n-1)(n-2)

Colby Messinger answered 2 years ago

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