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?
To find the number of different color patterns the company can produce, we can use the concept of permutations.
Since the stripes of equal width can have different colors, we need to count how many ways we can select colors for each stripe without considering the order.
Let's break down the problem step-by-step:
1. Choose a color for the background: Since the background color is fixed, there is only 1 way to choose it.
2. Choose the colors for the stripes: Since the stripes can have any of the 12 available colors, we have 12 choices for the first stripe, 12 choices for the second stripe, and so on. Since the stripes have equal width, any order of the color choices will result in the same pattern. Therefore, we need to calculate the number of combinations rather than permutations.
The formula for the number of combinations is nCr = n! / (r! * (n-r)!), where n is the total number of choices, and r is the number of choices we want to make.
In our case, we have 12 colors to choose from, and we want to choose 12 stripes (with repetition allowed). Plugging in these values into the combination formula, we get:
12C12 = 12! / (12! * (12-12)!) = 1
So, there is only 1 combination possible for selecting the colors of the stripes.
3. Calculate the final number of patterns: Since the background color is chosen independently of the stripe colors, we need to multiply the number of background color choices (1) by the number of stripe color combinations (1).
1 * 1 = 1
Therefore, the company can produce 1 different color pattern if it stocks material in 12 colors.
To determine the number of different color patterns the company can produce, we need to understand the concept of permutations.
In this scenario, we need to select multiple colors from a set of 12 colors and arrange them in a specific order to create the jacket's color pattern. Here's how we can approach this problem:
1. Determine the number of choices for the background color:
Since the background color cannot be the same as any of the stripes, we have 12 options to choose from initially.
2. Determine the number of choices for the first stripe:
Once we choose the background color, we have 11 remaining colors to choose from for the first stripe.
3. Determine the number of choices for the second stripe:
After selecting the background color and the first stripe color, we have 10 remaining colors to choose from for the second stripe.
4. Continue this pattern until we reach the last stripe:
Following the same logic, the number of choices for each subsequent stripe will continue to decrease. For example, for the third stripe, we will have 9 options, and so on until we reach the last stripe.
5. Multiply the number of choices for each element:
To find the total number of different color patterns, we need to multiply the number of choices for each element. In this case, since we have the same number of color choices for each element, we multiply the number of choices at each stage:
12 (choices for the background color) x 11 (choices for the first stripe color) x 10 (choices for the second stripe color) x ... x 3 (choices for the last stripe color).
6. Calculate the result:
Using this approach, we can multiply the number of choices together to find the total number of different color patterns.
Therefore, the number of different color patterns the company can produce, given 12 available colors, can be calculated as:
12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 = 39,916,800
So, the company can produce 39,916,800 different color patterns using the given materials.