The chance of rolling 6 on the first attempt is ⅙. For first and second attempts it's 1/36, then 1/216, then 1/1296. The chance of no rolling a 6 on the 5th roll is ⅚(1/1296) and no 6 on the next two attempts=(125/216)(1/1296)=125/279936.
But the four 6s could come up in 4 out of 7 rolls=35, therefore the probability of rolling exactly 4 6s in 7 rolls of the die=4375/279936=0.01563 approx (1.563%).