I recently got this wrong on my homework because I interpreted the question incorrectly. I wanted to see how others might interpret it, and perhaps explain to me how to understand it differently.
The conditions: The Powerball lottery is decided every Wednesday and Saturday night by drawing five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls. The Powerball jackpot is won by matching all five white balls in any order and the red Powerball. Each ticket costs $2. A ticket is defined as a set of five white and 1 red number.
Question 1: How many possible different Powerball tickets can be purchased?
Question 2: How many possible different winning Powerball tickets are there?
My answers
1. I considered tickets with white numbers in different orders to be different tickets, so I calculated a permutation for white numbers (1,348,621,560) multiplied by the number of available red numbers (26). This gave me 35,064,160,560 different tickets that could be purchased.
2. I took this to mean how many possible combinations of white numbers and red numbers are there. So I calculated the combination of white numbers (11,238,513) and multiplied it by the available red numbers (26). This gave me 292,201,338 possible winning tickets.
Grading Notes: The notes said I should have used the answer to 2 for question 1. And that for question 2, there is only 1 winning ticket. I am trying to wrap my head around how to interpret what they actually wanted to know. I think I am thinking too hard about the semantics of the wording? I want to know where I am going wrong so I can avoid this in the future exams.
Best Answer
The key phrase is 'in any order'. This tells you that you are looking at combinations in which the order doesn't matter. If they describe one thing as 'in any order' such as the balls, they inherently describe the thing they match to, the tickets, as 'in any order'. When doing future questions, it's best to identify what can be in any order, what connects to it what doesn't. This allows you to identify whether you are calculating combinations or permutations, which from the best of my understanding, is what your question is asking.