Probability of Forming a Line on a 3x3 Grid with Random Selections

[atqamar]

Consider a 3-by-3 square grid. Suppose you pick 5 of the squares at random. What is the probability that at least 1 line of 3 squares is formed? (3 diagonal squares is NOT a line).

I do know that there are 6 combinations of lines that are possible. After that, I'm not sure how to follow along with this.

 

[EnumaElish]

number of possible lines/number of possible ways you can select 3 out of 9

 

[atqamar]

I considered that, but you are not choosing 3 squares at a time... but instead, you are choosing a group of 5 squares. And from these 5 squares, what is the probability that there is a "line of 3 squares".

 

[atqamar]

The fact that $$_9C_5=126$$ shows that there are a total of 126 possible ways a group of five squares can be selected at random. Now how many of these 126 arrangements of squares contain at least one "line of 3 squares"? Do I have to draw all the possibilities (and consider some of the possibilities can be rotated or reflected to get another possibility)?

So far I have found 49 of them (a total of 10 diagram arrangements which can occur multiple times with rotations or reflections). Should I continue this way, or is there a simpler way?

 

[Office_Shredder]

If the line goes up a side, top or bottom it's probably always a rotation of a form where it goes up the right

note you can only have one line. So you put the line up the right, you have 6 spaces for 2 squares, multiply by 4 for rotation. Then you have to consider if the line goes up the middle

 

[atqamar]

Thank you very much for the insight Office_Shredder. Now I can make only the possible arrangements, and the remaining arrangements of the 126 will be the ones that don't create a "line of 3 squares".