# In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winn

• Apr 10th 2008, 01:20 PM
help1
In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winn
In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winning numbers are drawn, then one supplemental number is drawn from the remaining numbers. You win Fifth Prize is the six numbers you have chosen contain exactly three of the winning numbers and the supplemental number. Find the probability of winning Fifth Prize.
• Apr 10th 2008, 04:25 PM
Soroban
Hello, help1!

Quote:

In a new lotto game, you must choose six numbers from the numbers 1 to 25.
Six winning numbers are drawn,
. . then one supplemental number is drawn from the remaining numbers.
You win Fifth Prize if your 6 numbers contain exactly 3 of the winning numbers
. . and the supplemental number.

Find the probability of winning Fifth Prize.

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

• Apr 10th 2008, 05:47 PM
help1
Quote:

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

What do those (25 6) mean? How do I show that = 177,100? What button on my calc does that?
• Apr 10th 2008, 05:58 PM
Mathstud28
I think it is
Quote:

Originally Posted by help1
What do those (25 6) mean? How do I show that = 177,100? What button on my calc does that?

A permutation so it would be equivalent to $\displaystyle \frac{25!}{6!(25-6)!}$ where ! denotes factorial...and you might have a button...it would look like this $\displaystyle _nP_r$