Probability question (grid)

Question

Say I have a grid of 10x7. Every square of that grid is empty. Then, 20 squares, chosen at random, are filled (a square can only be filled once, so no duplicates allowed). What is the probability of getting at least one entire row of filled squares (in other words, the probability of 10 of those 20 "fillings" landing on the same row of 10 squares)?

Answer 1

Hint: Let us first get it wrong. There are $\binom{70}{20}$ ways to select $20$ squares, all equally likely.

A choice is favourable if some row is filled. Which row? It can be picked in $\binom{7}{1}$ ways. And then the rest of the squares can be filled in $\binom{60}{10}$ ways. That gives a total of $\binom{7}{1}\binom{60}{10}$ favourables.

However, we have double-counted every arrangement in which $2$ rows are filled. There are $\binom{7}{2}$ of these. Adjust.