13. (4 pts) At an annual flower show, 6 different entries are to be arranged in a row.
a) How many different arrangements of the entries are possible?
b) If the owners of the 1st, 2nd, and 3rd place entries will be awarded prizes of $100, $50, and $25 respectively, how many ways can the prizes be awarded?
6 ways to put an entry in space 1 times
5 ways to put an entry in space 2 times
4 ways to put an entry in space 3 times
3 4
2 5
1 6
looks like 6! = 720
How many permutations of 6 taken 3 at a time? (They are permutations not combinations, because the order in the group of three matters)
n!/(n-r)!
where n is 6 and r is three
6!/(3)! = 6*5*4 = 120
Thank you Damon for all your help. You all are life savers.
a) To find the number of different arrangements of the entries, we can use the concept of permutations. In this case, we have 6 different entries to be arranged in a row. The number of permutations can be calculated using the formula:
nPr = n! / (n - r)!
Where n is the total number of options (6 entries) and r is the number of items to be arranged (also 6 in this case since we want to arrange all the entries).
So, for this question:
nPr = 6! / (6 - 6)!
= 6! / 0!
= 6! / 1
= 6 * 5 * 4 * 3 * 2 * 1 / 1
= 720
Therefore, there are 720 different arrangements of the entries possible.
b) Now, let's consider the second part of the question. We need to find the number of ways the prizes can be awarded to the entries. In this case, the order matters because the 1st, 2nd, and 3rd place prizes have different values.
To calculate the number of ways the prizes can be awarded, we can also use permutations. However, we need to select 3 positions out of the 6 available positions for the prizes.
The number of ways to select 3 positions from 6 can be calculated using the formula:
nCr = n! / (r! * (n - r)!)
Where n is the total number of positions (6) and r is the number of items to be selected (3).
So, for this question:
nCr = 6! / (3! * (6 - 3)!)
= 6! / (3! * 3!)
= 6! / (3 * 2 * 1 * 3 * 2 * 1)
= 6 * 5 * 4 / (3 * 2 * 1)
= 20
Therefore, there are 20 ways the prizes can be awarded.