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Chapter 1: Problem 14
Evaluate each of the following. a) \(P(7,2)\) b) \(P(8,4)\) c) \(P(10,7)\) d) \(P(12,3)\)
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a) Find the coefficient of \(v^{2} w^{4} x z\) in the expansion of \((3 v+2 w+x+y+z)^{8}\) b) How many distinct terms arise in the expansion in part (a)?
Consider the following program segment where \(i, j\), and \(k\), are integer variables. for \(i:=1\) to 12 do for \(j:=5\) to 10 do for \(k:=15\) downto \(8 \mathrm{do}\) print \((i-j) * k\) a) How many times is the print statement executed? b) Which counting principle is used in part (a)?
For any given set in a tennis tournament, opponent A can beat opponent \(\mathrm{B}\) in seven different ways. (At 6-6 they play a tie breaker.) The first opponent to win three sets wins the tournament. (a) In how many ways can scores be recorded with A winning in five sets? (b) In how many ways can scores be recorded with the tournament requiring at least four sets?
Write a computer program (or develop an algorithm) to list the integer solutions for a) \(x_{1}+x_{2}+x_{3}=10, \quad 0 \leq x_{l}, \quad 1 \leq i \leq 3\) b) \(x_{1}+x_{2}+x_{3}+x_{4}=4, \quad-2 \leq x_{i}, \quad 1 \leq i \leq 4\)
a) How many distinct paths are there from \((-1,2,0)\) to \((1,3,7)\) in Euclidean three-space if each move is one of the following types? \((\mathrm{H}):(x, y, z) \rightarrow(x+1, y, z)\) \((\mathrm{V}):(x, y, z) \rightarrow(x, y+1, z)\) \((\mathrm{A}):(x, y, z) \rightarrow(x, y, z+1)\) b) How many such paths are there from \((1,0,5)\) to \((8,1,7) ?\) c) Generalize the results in parts (a) and (b).
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