Name all the arrays you can make with each set of tiles.
1. 8 tiles
2. 9 tiles
3.16 tiles
8:
1x8
2x4
4x2
8x1
9:
1x9
3x3
9x1
I will leave you to attempt 16 as an exercise. Post your answer for checking if you wish.
1. With 8 tiles, you can create several arrays, such as:
a) A 2x4 array
b) A 4x2 array
c) A 1x8 array
d) An 8x1 array
2. With 9 tiles, you can create the following arrays:
a) A 3x3 square array
b) A 1x9 array
c) A 9x1 array
d) A 3x3 rectangular array
3. With 16 tiles, you can create numerous arrays, including:
a) A 4x4 square array
b) A 2x8 array
c) An 8x2 array
d) A 1x16 array
e) A 16x1 array
f) A 4x4 rectangular array
To determine all the arrays that can be made with each set of tiles, we need to consider the arrangements that can be formed by permuting the tiles within each set.
1. For 8 tiles:
To calculate the number of possible arrays, we can use the concept of permutations. Since all tiles are distinct (assuming no duplicates), we can think of arranging 8 tiles in a row. This can be represented as P(8, 8), which means the number of permutations of 8 objects taken all at once. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of objects and r is the number of objects to be arranged. In this case, it would be 8P8 = 8! / (8-8)! = 8! / 0! = 8!. Therefore, there are 8! = 8*7*6*5*4*3*2*1 = 40,320 different arrays that can be made with 8 tiles.
2. For 9 tiles:
Similarly, we calculate the number of permutations for 9 tiles using the formula. 9P9 = 9! / (9-9)! = 9! / 0! = 9!. Thus, there are 9! = 9*8*7*6*5*4*3*2*1 = 362,880 possible arrays with 9 tiles.
3. For 16 tiles:
Following the same logic, we have 16P16 = 16! / (16-16)! = 16! / 0! = 16!. The number of arrays here would be 16! = 16*15*14*...*3*2*1 = 20,922,789,888,000 possible combinations.
So, the answer is:
1. 8 tiles: 40,320 arrays
2. 9 tiles: 362,880 arrays
3. 16 tiles: 20,922,789,888,000 arrays