91Ó°ÊÓ

. If \(A=\\{1,2,3,4,5\\}\) and there are 6720 injective functions \(f: A \rightarrow B\), what is \(|B| ?\)

For \(m, n \in \mathbf{Z}^{+}\), prove that if \(m\) pigeons occupy \(n\) pigeonholes, then at least one pigeonhole has \(\lfloor(m-1) / n\rfloor+1\) or more pigeons roosting in it.

For the access function developed in Example 5.10(d), the matrix \(A=\left(a_{i j}\right)_{m \times n}\) was stored in a one-dimensional array using the row major implementation. It is also possible to store this matrix using the column major implementation, where each entry \(a_{11}, 1 \leq i \leq m\), in the first column of \(A\) is stored in locations \(1,2,3, \ldots, m\), respectively, of the array, when \(a_{11}\) is stored in location 1. Then the entries \(a_{12}, 1 \leq i \leq m\), of the second column of \(A\) are stored in locations \(m+1, m+2, m+3, \ldots, 2 m\), respectively, of the array, and so on. Find a formula for the access function \(g\left(a_{i j}\right)\) under these conditions.