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?
a) 6x5x4x3x2x1 = 6! = ?
b) Regardless of the dollar amount of the prizes, the same three people can share them 3! = 6 ways
a) To find the number of different arrangements of the entries, we use the concept of permutations. Since there are 6 different entries, we have 6 choices for the first spot, 5 choices for the second spot, 4 choices for the third spot, and so on until only 1 choice is left for the last spot. So the number of different arrangements can be found by multiplying these choices together:
Number of arrangements = 6 x 5 x 4 x 3 x 2 x 1 = 6!
Therefore, there are 6! = 720 different arrangements of the entries.
b) To find the number of ways the prizes can be awarded to the owners of the 1st, 2nd, and 3rd place entries, we can use the concept of permutations again. Since there are 3 people and 3 prizes, we have 3 choices for the first prize, 2 choices for the second prize, and 1 choice for the last prize. So the number of ways the prizes can be awarded can be found by multiplying these choices together:
Number of ways to award prizes = 3 x 2 x 1 = 3!
Therefore, there are 3! = 6 ways the prizes can be awarded.