50 people purchase raffle tickets 3 winning tickets are selected at random if the first prize is $1,000, second prize is $500, and third prize is $100 in how many different ways can the prizes be awarded and is this a permutations or a combinations problem?
Well, we're dealing with a permutations problem here! To determine the number of ways the prizes can be awarded, we need to multiply the number of choices for each prize.
For the first prize, there are 50 people competing, so there are 50 choices.
For the second prize, after the first prize is awarded, there are 49 people left, so there are 49 choices.
And for the third prize, after the first and second prizes are awarded, there are 48 people remaining, so there are 48 choices.
To calculate the total number of ways the prizes can be awarded, we multiply these choices together:
50 choices (for the first prize) × 49 choices (for the second prize) × 48 choices (for the third prize) = 117,600 different ways.
So, there are 117,600 different ways to award the prizes!
This problem involves determining the number of ways the prizes can be awarded, which indicates that it is a permutations problem.
To solve this, we need to find the number of permutations when selecting 3 winners out of 50 people.
The formula to find the number of permutations is: nPr = n! / (n - r)!, where n is the total number of options and r is the number of selections.
In this case, n = 50 (since there are 50 people purchasing tickets), and r = 3 (since we are selecting 3 winners).
Using the formula, we can calculate the number of ways the prizes can be awarded as follows:
50P3 = 50! / (50 - 3)!
= 50! / 47!
= (50 * 49 * 48 * 47!) / 47!
= 50 * 49 * 48
= 117,600
Therefore, there are 117,600 different ways the prizes can be awarded.
To determine the number of different ways the prizes can be awarded, we need to identify whether this problem involves permutations or combinations.
Permutations deal with arranging objects in a specific order, while combinations only consider the selection of objects without regard to order.
In this scenario, the order in which the prizes are awarded is significant since the prizes have different values. Therefore, we are dealing with permutations.
To calculate the number of permutations, we can use the formula for permutations of n objects taken r at a time:
P(n, r) = n! / (n - r)!
Where n is the total number of objects (in this case, 50 people) and r is the number of objects taken at a time (in this case, 3 prizes).
Now, let's substitute the values into the formula:
P(50, 3) = 50! / (50 - 3)!
Note that the exclamation mark represents the factorial operation, which means multiplying a series of descending numbers.
Calculating this expression will provide us with the total number of different ways the prizes can be awarded.
clearly a permutations problem, since the order of award matters.
So, there are 50P3 ways to arrange the selections.