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Chapter 1: Problem 14
How many 5 -card poker hands are there?
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Consider \(n\)-digit numbers where each digit is one of the 10 integers 0 , \(1, \ldots, 9 .\) How many such numbers are there for which (a) no two consecutive digits are equal; (b) 0 appears as a digit a total of \(i\) times, \(i=0, \ldots, n\) ?
Let \(H_{k}(n)\) be the number of vectors \(x_{1}, \ldots, x_{k}\) for which each \(x_{i}\) is a positive integer satisfying \(1 \leq x_{i} \leq n\) and \(x_{1} \leq x_{2} \leq \cdots \leq x_{k}\) (a) Without any computations, argue that $$ \begin{aligned} &H_{1}(n)=n \\ &H_{k}(n)=\sum_{j=1}^{n} H_{k-1}(j) \quad k>1 \end{aligned} $$ HINT: How many vectors are there in which \(x_{k}=j\) ? (b) Use the preceding recursion to compute \(H_{3}(5)\). HINT: First compute \(H_{2}(n)\) for \(n=1,2,3,4,5\).
If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each school must receive 2 teachers?
Consider a function \(f\left(x_{1}, \ldots, x_{n}\right)\) of \(n\) variables. How many different partial derivatives of order \(r\) does it possess?
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 ; the third digit was any integer between 1 and 9. How many area codes were possible? How many area codes starting with a 4 were possible?
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