isi-entrance 2007 Q8

isi-entrance · India · solved Permutations & Arrangements Permutation Properties and Enumeration (Abstract)

In how many ways can the numbers $1, 2, \ldots, 9$ be arranged in a $3 \times 3$ grid \[ \begin{array}{|c|c|c|} \hline A & B & C \\ \hline D & E & F \\ \hline G & H & I \\ \hline \end{array} \] such that each row and each column is in increasing order (i.e., $A < B < C$, $D < E < F$, $G < H < I$, $A < D < G$, $B < E < H$, $C < F < I$)?

$G \times D \times A \times H \times I \times E \times B \times F \times C = 9 \times 6 \times 3 \times 6 \times 3 \times 4 \times 2 \times 2 \times 1$ ways.

In how many ways can the numbers $1, 2, \ldots, 9$ be arranged in a $3 \times 3$ grid
\[
\begin{array}{|c|c|c|}
\hline
A & B & C \\
\hline
D & E & F \\
\hline
G & H & I \\
\hline
\end{array}
\]
such that each row and each column is in increasing order (i.e., $A < B < C$, $D < E < F$, $G < H < I$, $A < D < G$, $B < E < H$, $C < F < I$)?

Paper Questions

Q1 Q3 Q5 Q7 Q8 Q10