There are 5 people in a raffle drawing. Two raffle winners each win gift cards. Each gift card is the same. How many ways are there to choose the winners?
Decide if the situation involves a permutation or combination, and then find the number of ways to choose the winners.
A. Permutation; number of ways = 10
B. Combination; number of ways = 20
C. Combination; number of ways = 10
D. Permutation; number of ways = 20
Answer (2)
The situation involves a combination since the order of winners does not matter. Using the combination formula, we find that there are 10 ways to choose the winners. Therefore, the answer is C. Combination; number of ways = 10.
;
Determine that order doesn't matter, so it's a combination.
Apply the combination formula: C ( n , k ) = k ! ( n − k )! n ! .
Calculate C ( 5 , 2 ) = 2 ! 3 ! 5 ! = 10 .
Conclude there are 10 ways to choose the winners.
Explanation
Identify the problem type We have 5 people in a raffle, and we want to choose 2 winners. Since the gift cards are identical, the order in which we choose the winners doesn't matter. This means we're dealing with a combination problem, not a permutation problem.
State the combination formula The combination formula is given by: C ( n , k ) = k ! ( n − k )! n ! where n is the total number of items, and k is the number of items we want to choose. In our case, n = 5 (the number of people) and k = 2 (the number of winners).
Calculate the number of combinations Now, let's plug in the values and calculate the number of combinations: C ( 5 , 2 ) = 2 ! ( 5 − 2 )! 5 ! = 2 ! 3 ! 5 ! = ( 2 × 1 ) ( 3 × 2 × 1 ) 5 × 4 × 3 × 2 × 1 = 2 × 1 5 × 4 = 2 20 = 10 So, there are 10 ways to choose the two winners.
Final Answer Since the order of winners does not matter, this is a combination problem. We calculated that there are 10 possible ways to choose the two winners from the five people.
Examples
Consider a scenario where a teacher wants to select a group of students for a project. If the teacher needs to pick 2 students out of a class of 5, and all students have the same role in the project, the order of selection doesn't matter. This is a combination problem, and the teacher can choose the students in 10 different ways. Understanding combinations helps in scenarios like team selections, lottery drawings, and any situation where the order of selection is not important.
