Let f(k, m) be the number of k-dimensional faces of the m-cube. See Table 1. k=0...

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Let f(k, m) be the number of k-dimensional faces of the m-cube. See Table 1. k=0 k = 1 k = 2 k 3 k = 4 m = 1 m = 2 4 4 1 0 0 2 1 0 0 0 m = 3 8 12 6 1 0 m = 4 f(0,4) f(1,4) ƒ(2,4) f(3, 4) ƒ(4,4) m=5 f(0,5) f(1,5) ƒ(2,5) f(3,5) f(4,5) ... m f(0, m) f(1, m) f(2, m) f(3, m) f(4, m) m + 1 f(0, m + 1) f(1, m+1) f(2, m + 1) f(3, m + 1) f(4, m + 1) Table 1 f(k, m) is the number of k-dimensional faces of the m-cube. (a) Verify the numbers in the first three columns. (b) Calculate the columns m = 4, m = 5, and give the formula for passing from the mth column to the (m + 1)st. (c) What would an m = 0 column mean? (d) Prove that the alternating sum of the entries in any column is 1. That is, 2−1=1, 4−4+1=1, 8−12+6−1 = 1, and in general Σ(−1)k f(k, m) = 1. This alternating sum is called the Euler characteristic. Let f(k, m) be the number of k-dimensional faces of the m-cube. See Table 1. k=0 k = 1 k = 2 k 3 k = 4 m = 1 m = 2 4 4 1 0 0 2 1 0 0 0 m = 3 8 12 6 1 0 m = 4 f(0,4) f(1,4) ƒ(2,4) f(3, 4) ƒ(4,4) m=5 f(0,5) f(1,5) ƒ(2,5) f(3,5) f(4,5) ... m f(0, m) f(1, m) f(2, m) f(3, m) f(4, m) m + 1 f(0, m + 1) f(1, m+1) f(2, m + 1) f(3, m + 1) f(4, m + 1) Table 1 f(k, m) is the number of k-dimensional faces of the m-cube. (a) Verify the numbers in the first three columns. (b) Calculate the columns m = 4, m = 5, and give the formula for passing from the mth column to the (m + 1)st. (c) What would an m = 0 column mean? (d) Prove that the alternating sum of the entries in any column is 1. That is, 2−1=1, 4−4+1=1, 8−12+6−1 = 1, and in general Σ(−1)k f(k, m) = 1. This alternating sum is called the Euler characteristic.

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ANSWER a The first three columns of the table can be easily verified f01 2 since a 1cube line segmen... View the full answer