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Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in \( 30 \) states, Washington D.C., and the U.S. Virgin Islands.The game is played by drawing five white balls out of a drum of \( 59 \) white balls (numbered \( 1 - 59 \)) and one red powerball out of a drum of \( 39 \) red balls (numbered \( 1 - 39 \)). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers if the jackpot is won by matching all five white balls in order and the red power ball. (c) Compare the results of part (a) with a state lottery in which a jackpot is won by matching six balls from a drum of \( 59 \) balls.

Step by step solution

01

Winning Powerball lottery in any order

For this part, use the combination formula, where \( C(n,k) = n! / k!(n-k)! \). Choose 5 white balls out of 59: \( C(59,5) \). Then select 1 red ball out of 39: \( C(39,1) \). Multiply these two together to get the total possibilities. The '!' denote factorial operation, whereby \( n! = n × (n−1) × (n−2) × ... × 3 × 2 × 1 \).

02

Winning Powerball lottery in order

For this part, use the permutations formula, where \( P(n,k)=n!/(n-k)! \). Choose 5 white balls out of 59 and arrange them: \( P(59,5) \). Then select 1 red ball out of 39: \( C(39,1) \). Multiply these two together to get the total possibilities.

03

Winning a state lottery

Using the combination formula again, evaluate how many ways there are to choose 6 balls from 59: \( C(59,6) \). This will give the total possibilities for the state lottery.

04

Compare Results

Comparing the results of part (a) and (c), this will show the odds of winning a Powerball lottery (with and without considering the order of the white balls) against a standard state lottery.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combination

When considering combinations, the order of selection does not matter. This is crucial in many lottery-style problems, such as our Powerball game where we care only about the chosen numbers, not the sequence.

The formula to calculate combinations is given by:\[ C(n,k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of available items (or digits), and \( k \) is the number of items to choose.

For example, if we're choosing 5 white balls from a total of 59, we calculate it as \( C(59,5) \), resulting in the total number of ways to choose and not care about the order.

• Choose 5 white balls from 59: Calculate \( C(59,5) \).
• Once the white balls are selected, choose 1 from 39 red balls: Calculate \( C(39,1) \).
• Multiply both to get the total combination of selections.

Combinations are standard when dealing with lotteries where sequence isn't important but selection is.

Permutation

Unlike combinations, permutations concern themselves with order, meaning how items are arranged matters. This occurs in cases where particular sequences of numbers are relevant, such as when Powerball specifically requires balls to be in a particular sequence.

The permutation formula is defined as: \[ P(n,k) = \frac{n!}{(n-k)!} \] where again \( n \) is the total count and \( k \) is the number of selections. Here, the arrangement is key.

In the Powerball example, if we're required to arrange 5 white balls, we compute it through \( P(59,5) \).

• Permute 5 white balls from 59: Calculate \( P(59,5) \).
• Then select 1 from 39 red balls: Calculate \( C(39,1) \) (since order of a single item does not matter).
• The total permutations are found by multiplying these outcomes.

Permutations thus add complexity and impact results notably, differing from the orderless selection seen with combinations.

Factorial

Factorials are integral to both combinations and permutations, as they facilitate calculations of total possible arrangements or selections. The factorial of a number, denoted \( n! \), is the product of all positive integers up to \( n \).

For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factoring plays a vital role in reducing complex arithmetic into more manageable operations when dealing with large numbers, typical in lottery calculations.

In permutation and combination formulas:

• Used in the denominator to divide out unnecessary arrangements, focusing calculations on valid selections only.
• Aids in structuring both combination \( C(n, k) \) and permutation \( P(n, k) \) equations, simplifying their use.

Without factorials, calculating the odds in games like Powerball would be cumbersome, due to the astronomically high number of possible outcomes involved.