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Chapter 16: Problem 7

a) In how many ways can we paint the eight squares of a \(2 \times 4\) chessboard, using the colors red and white? (The back of the chessboard is black cardboard.) b) Find the pattern inventory for the colorings in part (a). c) How many of the colorings in part (a) have four red and four white squares? How many have six red and two white squares?

Short Answer

Expert verified
a) There are \(256\) ways of painting the chess board. b) The pattern is represented by \(A^4B^4\), where \(A\) represents a red square and \(B\) represents a white square. c) There are \(70\) combinations of 4 red squares and 4 white squares, and \(28\) combinations of 6 red squares and 2 white squares.

Step by step solution

01

Calculate total number of ways

There are 8 squares on the chessboard. For each square, there are 2 choices for what color can be used to paint it. To calculate the total number of ways to paint the board, multiply the number of choices for each square. Hence total combinations are \(2^8 = 256\) combinations.
02

Find the pattern inventory

As the problem doesn't request for the explicit demonstration of 256 combinations, but to find the pattern inventory, the representative pattern can be found by following the representative method of combinatoric. Because chessboard is \(2\times4\) and there are at most 3 contiguous squares in each row and column, the representative is \(A^4 B^4\), where \(A\) represents a red square and \(B\) represent a white square.
03

Count specific color combinations

In this case we're looking for 2 different combinations: exactly 4 red squares and 4 white squares, and exactly 6 red squares and 2 white squares. For both cases, remember that we're choosing spots on the chessboard for the squares to go, and the positions are distinct. For 4 red and 4 white, it's \(\binom{8}{4} = 70\) ways. For 6 red and 2 white, it's \(\binom{8}{2} = 28\) ways. Note that \(\binom{n}{k}\) represents a binomial coefficient, the number of ways to choose \(k\) unique items from \(n\) total items.

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Most popular questions from this chapter

If \(G\) is a group of even order, prove that there is an element \(a \in G\) with \(a \neq e\) and \(a=a^{-1}\).

a) Construct a decoding table (with syndromes) for the group code given by the generator matrix $$ G=\left[\begin{array}{lllll} 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \end{array}\right] $$ b) Use the table from part (a) to decode the following received words. \(\begin{array}{cccc}11110 & 11101 & 11011 & 10100 \\ 10011 & 10101 & 11111 & 01100\end{array}\) c) Does this code correct single errors in transmission?

For each of the following encoding functions, find the minimum distance between the code words. Discuss the errordetecting and error-correcting capabilities of each code. a) \(\begin{array}{rl}E: \mathbf{Z}_{2}^{2} \rightarrow \mathbf{Z}_{2}^{5} \\\ 00 \rightarrow 00001 & 01 \rightarrow 01010 \\ 10 \rightarrow 10100 & 11 \rightarrow 11111\end{array}\) b) \(E: \mathbf{Z}_{2}^{2} \rightarrow \mathbf{Z}_{2}^{10}\) \(\begin{array}{cl}00 \rightarrow 0000000000 & 01 \rightarrow 0000011111 \\ 10 \rightarrow 1111100000 & 11 \rightarrow 1111111111\end{array}\) \begin{array}{ll} \text { c) } E: \mathbf{Z}_{2}^{3} \rightarrow \mathbf{Z}_{2}^{6} & \\ 000 \rightarrow 000111 & 001 \rightarrow 001001 \\ 010 \rightarrow 010010 & 011 \rightarrow 011100 \\ 100 \rightarrow 100100 & 101 \rightarrow 101010 \\ 110 \rightarrow 110001 & 111 \rightarrow 111000 \\ \text { d) } E: \mathbf{Z}_{2}^{3} \rightarrow \mathbf{Z}_{2}^{8} & \\ 000 \rightarrow 00011111 & 001 \rightarrow 00111010 \\ 010 \rightarrow 01010101 & 011 \rightarrow 01110000 \\ 100 \rightarrow 10001101 & 101 \rightarrow 10101000 \\ 110 \rightarrow 11000100 & 111 \rightarrow 11100011 \end{array}

Let \(G\) be a group with subgroups \(H\) and \(K\). If \(|G|=660\), \(|K|=66\), and \(K \subset H \subset G\), what are the possible values for | \(H \mid ?\)

In \(S_{5}\) find an element of order \(n\), for all \(2 \leq n \leq 5\). Also determine the (cyclic) subgroup of \(S_{5}\) that each of these elements generates.
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