suppose we color the squares of a 4×4 grid blue or red.Using inclusion-exclusion, how many ways can we color it so that no row has only one color
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suppose we color the squares of a 4×4 grid blue or red.Using inclusion-exclusion, how many ways can we color it so that no row has only one color
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- Count the number of strings of m 0s and n 1s that have exactly k blocks of consecutive 1s.Below are the first five letters of the alphabet, designed to fit the vertices of a grid. Below them in subparts (a) through (d) are the first four letters that were designed on other grids. Draw the "e" that goes with each type style, or font. As stated many times in this book, we want you to go beyond random trial and error. First, take some time and think. One strategy is to try to write the rules that the writer of the letter used. Another strategy is to try to articulate the common characteristics of the four letters.t abcde (a) (b) (c) تحاد EEEE abcd. BA م داد plo EEEE 0In a town of 10000 families it was found that 40% of families buy newspaper A, 20% family buy newspaper B, 10% family buy newspaper C, 5% family buy newspaper A and B, 3% family buy newspaper B and C and 4% family buy newspaper A and C. If 2% family buy all the newspaper. Find the number of families which buy Number of families which buy all three newspapers. Number of families which buy None of A, B, C
- A monkey has filled in a 3 × 3 grid with the numbers 1, 2, . . . , 9. A cat writes down thethree numbers obtained by multiplying the numbers in each horizontal row. A dog writesdown the three numbers obtained by multiplying the numbers in each vertical column.Can the monkey fill in the grid in such a way that the cat and dog obtain the same listsof three numbers? What if the monkey writes the numbers 1, 2, . . . , 25 in a 5×5 grid? Or1, 2, . . . , 121 in a 11 × 11 grid? Can you find any conditions on n that guarantee that itis possible or any conditions that guarantee that it is impossible for the monkey to writethe numbers 1, 2, . . . , n^2 in an n × n grid so that the cat and the dog obtain the same lists of numbers? Please provide a mathematically rigorous solution with proofs that could be understood by a person at a high school level.2. Minimize z - 6х, + 6х, + 8x, + 9х, subject to X, + 2xz + x3+ X4 2 3 2x, + xz + 4x3 + 9x42 8 X, 2 0, x2 2 0, x3 2 0, x,2 0.in how many ways you can put dominos(1x2) on a grid(nxn)? Give at least solution for 4x4 and 6x6. I know there exact answer with conplecated fornula.
- Re6. When bees play chess, they use a hexagonal board like the one shown below. The queen bee can move one space at a time either directly to the right or angled up-right or down-right (but can never move leftwards). How many different paths can the queen take from the top left hexagon to the bottom right hexagon? Explain your answer, and this relates to the previous question. (As an example, there are three paths to get to the second hexagon on the bottom row.) ntart wtopA rectangular board, 3 by 2, is divided into 1 by 1 squares. Each square can be coloured red or black. (Note that if one colouring can be obtained from another by rotation, the two colourings are distinct.) If three squares are red and three are black, in how many ways can the board be coloured?
