Problem 54
Question
In Exercises \(49-58,\) solve by the method of your choice. Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?
Step-by-Step Solution
Verified Answer
The prizes could be awarded in 19600 different ways.
1Step 1: Understanding Combinations
Combination is used when order doesn't matter. If we have 'n' options and we need to choose 'r' of them, we can calculate the number of combinations using the formula \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where '!' denotes factorial, which means multiplying all positive integers up to that number. Here, we have 50 people who bought the tickets (n = 50) and 3 prizes to give away (r = 3).
2Step 2: Apply Combination Formula
Applying the values in the combination formula: \[ C(50, 3) = \frac{50!}{3!(50-3)!} \]\[ = \frac{50!}{3!47!} \] To simplify this, we don't need to calculate the entire value of 50!. We can simplify 50! as 50*49*48*47!. The 47! in the numerator and denominator will cancel out. So the expression becomes \[ = \frac{50*49*48}{3!} \] and 3! is 3*2*1 = 6.
3Step 3: Simplify the Expression
Finally, the expression simplifies to: \[ C(50, 3) = \frac{50*49*48}{6} \] After calculating this expression, we get the total number of ways the prizes can be awarded.
Key Concepts
FactorialsProbability theoryPermutations and combinations
Factorials
Factorials play a crucial role in calculating combinations. In simple terms, the factorial of a number, denoted by an exclamation mark (e.g., \( n! \)), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials grow very fast with increasing numbers. As such, they are useful for counting permutations and combinations. However, because they grow large quickly, it's often more efficient to simplify them when possible, especially if they appear in both the numerator and denominator of a fraction, as seen in our original exercise.
It's important to remember the special case: \( 0! = 1 \), even though multiplying by zero typically results in zero. This is a defined rule to make calculations involving factorials consistent.
Factorials grow very fast with increasing numbers. As such, they are useful for counting permutations and combinations. However, because they grow large quickly, it's often more efficient to simplify them when possible, especially if they appear in both the numerator and denominator of a fraction, as seen in our original exercise.
It's important to remember the special case: \( 0! = 1 \), even though multiplying by zero typically results in zero. This is a defined rule to make calculations involving factorials consistent.
Probability theory
Probability theory helps us understand and quantify uncertainty. It deals with the likelihood of different outcomes in random events, such as flipping a coin or drawing a card from a deck.
The fundamental building block of probability is the probability of an event, which is a value between 0 and 1, where 0 means the event will not happen, and 1 means it will definitely happen. The basic formula for probability is:
Probability theory forms the basis for understanding how combinations and permutations work, as these concepts determine the number of possible arrangements or selections.
The fundamental building block of probability is the probability of an event, which is a value between 0 and 1, where 0 means the event will not happen, and 1 means it will definitely happen. The basic formula for probability is:
- \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)
Probability theory forms the basis for understanding how combinations and permutations work, as these concepts determine the number of possible arrangements or selections.
Permutations and combinations
Permutations and combinations are ways to arrange and select items, which is key when dealing with problems in probability and counting.
**Permutations** refer to arrangements where order matters. For example, in a race, coming first, second, or third is different from coming third, first, second. The formula for permutations of \( n \) items taken \( r \) at a time is
\[ P(n, r) = \frac{n!}{(n-r)!} \]
**Combinations**, on the other hand, deal with selections where order doesn't matter. Consider a team of three people selected from a group of ten. The order of selection doesn't change the team.
**Permutations** refer to arrangements where order matters. For example, in a race, coming first, second, or third is different from coming third, first, second. The formula for permutations of \( n \) items taken \( r \) at a time is
\[ P(n, r) = \frac{n!}{(n-r)!} \]
**Combinations**, on the other hand, deal with selections where order doesn't matter. Consider a team of three people selected from a group of ten. The order of selection doesn't change the team.
- The result for combinations is calculated by the formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Other exercises in this chapter
Problem 54
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