Problem 18 - Algebra and Trigonometry

Question

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000,\) second prize is \(\$ 500,\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

Step-by-Step Solution

Answer

There are 117,600 different ways the prizes can be awarded.

Step 1: Identify the given values

There are fifty people who've purchased raffle tickets - so, n = 50. Three tickets are to be selected as winners - first, second and third, hence r = 3.

Step 2: Apply the permutation formula

We simply plug in the values of n and r into the permutation formula \(nPr = n! / (n-r)!\), which yields 50P3 = 50! / (50 - 3)!.

Step 3: Calculate the result

This simplifies to 50! / 47!, which can further be simplified as 50 * 49 * 48. Calculate that product and obtain the result.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns in sets of objects. In the context of our raffle ticket problem, combinatorics helps us determine the number of possible ways to award prizes. When order matters in the arrangement, as it does with the raffle prizes (since the prizes are of different values - $1000, $500, $100), we use permutations.
In this problem, we use permutations because we want to calculate different ways to assign three distinct prizes to three different winners from a group of fifty people.

• Order is important: Picking a person for the first prize is different from picking the same person for the second or third prize.
• Permutations are used because each unique sequence of winners represents a different arrangement.

This is distinct from combinations, where the order does not matter. Thus, combinatorics involves choosing the right method to solve the problem – in this case, permutations.

Probability Theory

Probability theory is the study of randomness and uncertainty. It involves the analysis of random events and the calculation of the likelihood of these events. In a raffle draw, probability theory helps quantify how likely it is for any particular outcome to occur.
However, in this specific exercise, while the problem is focused more on counting (hence combinatorics and permutations), the understanding of probability remains relevant. Typically, one might wish to know the probability of any single ticket winning a prize. Still, here we are tasked with counting all possible outcomes.

• Calculation of odds: If interested in the probability, you would divide the number of ways a specific outcome can occur (like winning any prize) by the total number of outcomes.
• Application in real-world scenarios: Classic probability questions often involve noting an outcome's likelihood, like drawing a winning ticket.

While probability is not directly calculated, understanding this theory provides a basis for understanding permutations when order influences outcomes.

Factorials

Factorials are a fundamental component in permutations. A factorial of a number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). Factorials grow very rapidly and are used to compute permutations and combinations effectively.
In the raffle problem, the formula for permutations \( nP_r = \frac{n!}{(n-r)!} \) leverages factorials to determine how many different ways the prizes can be distributed. This is why we compute \( 50! \), dividing it by \( (50-3)! \) to simplify computations. Here, the factorial calculation looks like this:

• \( 50! = 50 \times 49 \times 48 \times \cdots \times 1 \), which is impractical to calculate fully.
• Since \( (50-3)! = 47! \), many terms cancel out, simplifying to \( 50 \times 49 \times 48 \).

Understanding how to manipulate and simplify factorials makes permutation problems, like the raffle, manageable and straightforward, showcasing their significant utility in combinatorics.