I tried to calculate a probability problem..pls review my try?

I will be straight and simple on explanation:

Abstract: I was playing bingo in a mobile game with my friend, and I wondered, "what is the probability of not getting a bingo?"
Objective: to calculate probablity of not getting a bingo.
* 5x5 board = 25 boxes total
* only number 1-25 are used
* no diagonal, only vertically and horizontally
* I hold a written(numbers1~25) board, and only my friend("speaker") who does NOT see the board calls out twenty(20) numbers very RANDOMLY (1~25).
Note: I do experience that I seldom get a game with no bingo. What is the probability?

My try (I am English literature student, so pls don't laugh...):
I looked up on internet threads...
- 25C20 means a probability of picking 20 random numbers out of 25 candidates.
- 25C20 = 53130
- 5 numbers shall not be marked in a row or column. This means..25x24x23x22x0..? well, this was the point where my head has blown up (already!??) and decided to ask for help.

Pls teach me the beauty of mathmatics!!! is it possible to calculate such odd?


Ambassador to the humans
Although there might be a simpler approach if you're really just interested in that one specific case of calling out 20 numbers out of 25.
how so..? something like having fixed number board?
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25

anyways, I have tried all night and my friend suggested 5!/25C20 (he just saw something like that on internet, we don't understand what it really means), could it be an answer?