A table showing the first thirteen rows of Pascal's triangle, where the rows are, as usual numbered from 0 to 12 is presented. The entries in the table are called binomial coefficients. In this note, the authors systematically delete rows from Pascal's triangle and, by trial and error, try to find a formula that allows them to add new rows to the table directly. First they skip one row, then two rows, then three rows, etc. Eventually they arrive at a famous relation known as the Vandermonde convolution. This material can be used as an unusual exercise whenever Pascal's triangle or the binomial theorem is introduced. It provides the students with the opportunity to explore number patterns and conjecture mathematical formulas. (Contains 7 tables.)