Originally Posted by

**Soroban** Hello, help1!

There are: .$\displaystyle {25\choose6}\:=\:177,100$ ways to choose the 6 numbers

. . and $\displaystyle 19$ choices for the supplemental number.

Hence, there are: .$\displaystyle 177,100 \times 19 \:=\:{\color{blue}3,364,900}$ possible outcomes.

To win Fifth Prize:

You must have 3 of the 6 winning numbers: .$\displaystyle {6\choose3} \:=\:20$ ways.

You must have the supplemental number: .$\displaystyle 1$ way.

Your other two numbers are from the 18 "losers": .$\displaystyle {18\choose2}\:=\:153$ ways.

Hence, you have: .$\displaystyle 20 \times 1 \times 153 \:=\:{\color{blue}3,060}$ ways of winning.

Therefore: .$\displaystyle P(\text{5th Prize}) \;=\;\frac{3,060}{3,364,900} \;=\;\frac{153}{168,245}$