Lottery question

Hey guys, new here.

Just wondering if i am playing 23 games in a lottery, and IN TOTAL across the 23 games, the number 1 is drawn 9 times, what is the probability of that occurring?

some more info about this specific game, there are 45 balls, and each game consists of 7 balls/numbers, ie there have been a total of 161 numbers randomly chosen.

I'm thinking each 'game' needs to be thought of as independent, and following a hypergeometric distribution ie sampling without replacement. would the prob of drawing the number 1 in any one game be 7/45?

could I then think of it as a binomial dist where P(get a 1 in a game) = 7/45, and P(dont get 1 in a game) = 38/45, so P(get 9 1's in 23 games) = (23 choose 9) x (7/45)^9 x (38/45)^14 = 0.0041.

What do you guys think?