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Chapter 1: Q. 1.22 (page 19)
Consider a function of variables. How many different partial derivatives of order does possess?
The different partial derivatives of order that possess arelocalid="1648374000028" .
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Consider the grid of points shown at the top of the next column. Suppose that, starting at the point labelled A, you can go one step up or one step to the right at each move. This procedure is continued until the point labelled B is reached. How many different paths from A to B are possible? Hint: Note that to reach B from A, you must take steps to the right and steps upward.
The game of bridge is played by players, each of who is dealt cards. How many bridge deals are possible?
Consider a tournament of contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let localid="1648231792067" denote the number of different possible outcomes. For instance, localid="1648231796484" , since, in a tournament with localid="1648231802600" contestants, player localid="1648231807229" could be uniquely first, player localid="1648231812796" could be uniquely first, or they could tie for first.
(a) List all the possible outcomes when .
(b) With localid="1648231819245" defined to equal localid="1648231826690" , argue without any computations, that localid="1648281124813"
Hint: How many outcomes are there in which localid="1648231837145" players tie for last place?
(c) Show that the formula of part (b) is equivalent to the following:
localid="1648285265701"
(d) Use the recursion to find N(3) and N(4).
For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and 9, the second digit was either 0 or 1, and the third digit was any integer from 1 to 9. How many area codes were possible? How many area codes starting with a 4 were possible
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