You are thinking of using some leftover ceramic tiles to make a decorative border for a patio. You have eight blue, four gold, and nine white tiles.
In how many ways can you make the border if you must use all the tiles to put a single row along the edge of the patio? Show how you arrive at your answer.
To find the number of ways to make the border using all the tiles, we need to consider the arrangement of the tiles.
Since each row along the edge of the patio should contain all the tiles, the number of ways to arrange the tiles is equal to the number of permutations of the tiles.
To calculate the number of permutations, we can follow these steps:
Step 1: Calculate the total number of tiles.
Total number of tiles = number of blue tiles + number of gold tiles + number of white tiles
Total number of tiles = 8 + 4 + 9 = 21
Step 2: Calculate the factorial of the total number of tiles.
Factorial of a number (denoted as n!) is the product of all positive integers from 1 to that number.
To calculate the factorial of 21, multiply all positive integers from 1 to 21.
21! = 21 × 20 × 19 × 18 × ... × 1
Using a calculator or a computer program, the factorial of 21 can be calculated as:
21! ≈ 51,090,942,171,709,440,000
Therefore, there are approximately 51,090,942,171,709,440,000 ways to arrange the tiles to make a border.
Please note that this is an extremely large number, and it's not practical to list out all the possible arrangements.