Lucky six

Our strategy for solving this problem could be to first calculate the number of winning combinations, then to calculate the general number of combinations of picking $35$ balls out of $48$ , and finally to divide those two numbers.

We'll start with calculating the number of winning combinations. Winning combinations are the combinations of $35$ numbers which contain your six lucky numbers, while the other $29$ numbers can be any of $42$ numbers left. Hence, the number of winning combinations is equal to the number of ways of picking $29$ numbers out of $42$ , and that's:

$\binom{42}{29}=\frac{42!}{29!\times(42-29)!}=\frac{42!}{29!\times 13!}$

Next, number of combinations of picking $35$ balls out of $48$ is:

$\binom{48}{35}=\frac{48!}{35!\times(48-35)!}=\frac{48!}{35!\times 13!}$

Probability of winning is the quotient of these two values:

$\frac{\frac{42!}{29!\times 13!}}{\frac{48!}{35!\times 13 !}}=\frac{35\times 34\times 33\times 32\times 31\times 30}{48\times 47\times 46\times 45\times 44\times 43}=0.13227058\approx 0.132$