Calculate probability of a filling scale

Hi!
I am trying to calculate probability of filling scale by some points till to several different levels in a slot game during certain number of spins.
If defined symbol was triggered in a spin then scale get one ponit. For example the scale has three levels (r1=6,r2= 9, r3=12).
My task is to calculate the probability to reach each level. After reaching specific level, a small game will be started. I need probability of starting this small game.
This is what i am doing using Bernulli equation:
p - probability of appearing the trigger symbol on one place on the screen in a certain number of spins.
k = 3*5 = 15 - size of a slot's screen.
n - certain number of spins
r1=6,r2= 9, r3=12 - defined levels on the scale
nk = n*k - number of events of dropping symbols during n spins (one screen in one spin contains k = 15 places for symbols)
P1 - probability of the reaching level r1 = 6 points in n spins

P1 = nk!/(r1!*(nk - r1)!)*p^r1*(1-p)^(nk-r1)

Is it correct way to estimate that probability P1?
Thank you!