How will the probability change if I choose two, or three numbers?

I was thinking: there is a 1/12 chance that the first ball i pick could be the number I selected (i.e. 5) and 1/11 chance in the next ball...etc. so if i only chose one number, then my probability would be 1/12 x 1/11 x 1/10 x 1/9 = 1/11880

if i selected two numbers, then the probability would be 2/12 x 2/11 x 2/10 x 2/9 = 2/1485.

I tried to improve my chances by selecting more numbers. However, if I decided to improve my chance another way- by picking out more balls. Then the formula will fail because- I choose one number (ie. 5) and I pick five balls, 1/12 x 1/11 x 1/10 x 1/9 x 1/8 = 1/95040! The probability actually decreases if i try to improve my chances.

I need help with this.

It would be of great help if someone could detail the variations in the probabilities with changes to the number of numbers selected at the beginning. THANKS SO MUCH.