Probability, combinations, and permutations

Question

The question is based off the powerball lottery where you must get 5 numbers correctly followed by a powerball number in order to win the grand prize. The 5 numbers are done without replacement and must be within the number range 1-69 and the powerball number must be within range of 1-26.

https://www.powerball.com/games/powerball if you look at the powerball website, you must match 1 combination and a powerball or just the powerbal to win the 4 dollar prize.

The question is how many configurations win the 4 dollar prize?

Answer 1

The $4 prize can be won by matching these numbers in 2 different ways:

1. one out of five numbers (say, normal number) + powerball number

Those five numbers can be chosen in (69 choose 5) = 11238513 different ways

Now, the normal number, can match with any one of these 5 numbers in 5 different ways and

the powerball number can be chosen in 26 different ways.

So, there are 5⋅26⋅(69 choose 5) different ways of getting the $4 prize in this category.

2. powerball number

In this category, there is no match for the normal number(s).

So, there are only 26 different ways of getting $4 in this category.

Therefore, the $4 prize can be won in 26 + 5⋅26⋅(69 choose 5) different combinations.