i need help with this question.any help would be appreciated.
1.A company designs jackets that have stripes of equal width but different color.The background color of the jackets is different than the color of any of the stripes. How many different color patterns can the company prdouce if it stocks material in 12 colors
Assuming all stripes are the same color, 12 x 11 = 132
You multiply by 11 because, for each of twelve stripe colors, the background has to be one of the other 11 colors.
To solve this problem, we can use the concept of permutations.
The company designs jackets with stripes of equal width but different colors. We can think of the stripes as entities that need to be arranged in a specific order to form a pattern. In this case, the order matters because changing the order of the stripes will result in a different color pattern.
Since the company stocks material in 12 colors, there are 12 options for the first stripe. After selecting the first stripe color, there are 11 remaining colors available for the second stripe. Following the same pattern, there will be 10 options for the third stripe, 9 options for the fourth stripe, and so on.
To find the total number of possible color patterns, we need to multiply all the available options. Mathematically, this can be represented as:
12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
However, this cumbersome to calculate. Instead, we can express this using factorial notation. The factorial of a number is denoted by the symbol "!". It represents the product of all positive integers less than or equal to that number. For example, 5! (read as "5 factorial") is equal to 5 x 4 x 3 x 2 x 1.
In this case, the number of color patterns produced by the company can be calculated as:
12!
This means we need to calculate the factorial of 12. We can use a calculator or computer software to find the factorial value.
Therefore, the company can produce 12! (read as "12 factorial") different color patterns. Once you calculate the factorial value, you will have the answer to the question.