Number of tracingspaths across a rectangle (filled with letters) Given below is a words from the English dictionary arranged as a matrix MATHE ATHEM THEMA HEMAT EMATI MATIC ATICS Tracing the matrix is starting from the top left position and at each step move either RIGHT or DOWN, to reach the bottom right of the matrix. It is assured that any such tracing generates the same word. How many such tracings can be possible for a given word of length m+n-1 written as a matrix of size m * n?

Just to be consistent here, I'll assume that we're talking about $m$ rows and $n$ columns.

Each path then can be thought of as a Manhattan walk, where we have to go to the right $n-1$ times and down $m-1$ times. The total number of ways to do this is $$\binom{m+n-2}{n-1} = \binom{m+n-2}{m-1}$$ Think of it as either choosing the times we move right (left-hand side) or choosing the times we move down (right-hand side).