Problem 45
Show that there are \(\left(3^{n}+1\right) / 2\) strings of length \(n\) consisting of the letters \(a, b\), and \(x\) in which \(a\) occurs an even number of times.
Problem 45
A classroom has two rows of eight seats. There are 14 students in the class. Five students always sit in the front row, and four always sit in the back row. In how many ways can the students be seated?
Problem 46
How many ways are there to roll 10 dice so that all six different faces show?
Problem 46
For \(n=1,2,3, \ldots,\) write $$[x]_{t}=x(x-1)(x-2) \cdots(x-t+1)$$ for \(0 \leq t \leq n .\) We can represent \(\mid x]_{t}\) as a linear combination of powers of \(x .\) The coefficients for this expansion are denoted as \(s(n, t)\) and are known as the Stirling numbers of the first kind. Thus, for any \(n,\) we can write $$[x]_{t}=\sum_{t=0}^{n} s(n, t) x^{t}$$ The numbers \(s(n, t)\) can be defined as \(s(n, 0)=0\) for \(n=1,2,3, \ldots ; s(n, n)=1\) for \(n=0,1,2, \ldots ;\) and $$s(n, t)=s(n-1, t-1)-(n-1) s(n-1, t)$$ for \(t=1,2, \ldots, n-1 .\) Make a table of the Stirling numbers of the first kind for \(n=\) 1,2,3,4,5,6
Problem 47
For a positive integer \(t,\) define \([x]_{t}=x(x-1) \cdots(x-t+1) .\) We can represent \(x^{n}\) as a linear combination of \([x]_{t},\) where \(n=1,2,3, \ldots,\) and \(t=0,1,2, \ldots, n\). The coefficients for this expansion are denoted as \(S(n, t)\) and are known as the Stirling numbers of the second kind. Thus, for any \(n\), we can write $$x^{n}=\sum_{t=0}^{n} S(n, t)[x]_{t}$$ The numbers \(S(n, t)\) can be defined for \(n=1,2,3, \ldots\) as \(S(n, 0)=0 ; S(n, n)=1\) : and $$S(n, t)=t S(n-1, t)+S(n-1, t-1)$$ for \(1 \leq t \leq n-1\). Make a table of the Stirling numbers of the second kind for \(n=\) 1,2,3,4,5,6
