There are
[tex]\dbinom{51}6=\dfrac{51!}{6!(51-6)!}=18,009,460[/tex]
total ways to draw any 6 numbers from the range 1 to 51, regardless of order.
Given 6 selected numbers that match those drawn by the lottery, there are
[tex]6!=720[/tex]
ways of rearranging them. So the probability of winning 1st prize is
[tex]\dfrac{720}{18,009,460}=\dfrac{36}{900,473}\approx0.0000399790[/tex]
Next, given 6 selected numbers of which 5 match those drawn by the lottery, there are
[tex]5!=120[/tex]
ways of rearranging those 5 matching numbers. There are 46 remaining numbers that didn't get drawn, so the probability of winning 2nd prize is
[tex]\dfrac{46\cdot120}{18,009,460}=\dfrac{12}{39,151}\approx0.0003065[/tex]
