a) How many different ways can 50 players in a marching band be in rectangle arrangements?
b) If marching bands vary from 21 to 49 players, which number of players can be arranged in the greatest number of rectangles?
a) To determine the number of different ways 50 players in a marching band can be in rectangle arrangements, we need to find the factors of 50 and calculate all the possible combinations.
Step 1: Find the factors of 50.
The factors of 50 are: 1, 2, 5, 10, 25, and 50.
Step 2: Calculate the possible combinations.
To find the number of different ways, we need to calculate the number of pairs we can make using the factors. For each pair, we multiply the two factors to find the rectangle arrangements.
1 x 50 = 50 (1 row, 50 columns)
2 x 25 = 50 (2 rows, 25 columns)
5 x 10 = 50 (5 rows, 10 columns)
10 x 5 = 50 (10 rows, 5 columns)
25 x 2 = 50 (25 rows, 2 columns)
50 x 1 = 50 (50 rows, 1 column)
So, there are 6 different ways that 50 players in a marching band can be arranged in rectangle formations.
b) To determine which number of players can be arranged in the greatest number of rectangles within the given range (21 to 49), we need to calculate the factors for each number and count the combinations.
Let's calculate the combinations for each number of players within the given range.
For 21 players:
Factors of 21: 1, 3, 7, 21
Combinations: 1 x 21 = 21
For 22 players:
Factors of 22: 1, 2, 11, 22
Combinations: 1 x 22 = 22, 2 x 11 = 22
For 23 players:
Factors of 23: 1, 23
Combinations: 1 x 23 = 23
For 24 players:
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Combinations: 1 x 24 = 24, 2 x 12 = 24, 3 x 8 = 24, 4 x 6 = 24
Continue this process for all the remaining numbers in the range (25 to 49) and count the number of combinations for each number.
After calculating all the combinations, you can determine which number of players can be arranged in the greatest number of rectangles based on the count of combinations for each number.