we decided on a new logo for our brand of t-shirts.It was to be 3 stripes;(1) each stripe could be colored in 1 of 2 different colors,but we didn't want adjacent stripes to be the same color.how many different logos could we make.(2) how many more logos could we make if we used 3 colors instead of 2. (3) if we had used 4 stripes and 2 colors,how many more logos could we have made than using 3 stripes and 2 colors

To answer these questions, we can use the concept of permutations and combinations.

Question 1: How many different logos could you make with 3 stripes and 2 colors?

For the first stripe, you can choose any of the 2 colors. For the second stripe, you can choose any of the remaining 2 colors, and for the third stripe, you can choose any of the remaining 2 colors. Since you don't want adjacent stripes to be the same color, you need to subtract the cases where adjacent stripes are of the same color.

To calculate this, we can break it down into two cases:
Case 1: The first and second stripes have the same color.
In this case, there are 2 options for the first stripe, and only 1 option for the second stripe since it has to be the same color as the first stripe. For the third stripe, you have 2 color options. So, there are 2 * 1 * 2 = 4 possibilities.

Case 2: The first and second stripes have different colors.
In this case, there are 2 options for the first stripe and 2 options for the second stripe. For the third stripe, you have 2 color options. So, there are 2 * 2 * 2 = 8 possibilities.

To get the total number of possibilities, we add the possibilities from both cases: 4 + 8 = 12.

Therefore, you could make 12 different logos using 3 stripes and 2 colors without having adjacent stripes of the same color.

Question 2: How many more logos could you make if you used 3 colors instead of 2?

If you have 3 colors to choose from instead of 2, the number of possibilities for each stripe remains the same. The only difference is that you now have an additional color option for each stripe.

Using the same logic as before, there are 3 options for each stripe. Therefore, the total number of possibilities would be 3 * 3 * 3 = 27.

To calculate the additional logos you can make, subtract the number of logos with 2 colors (12) from the number of logos with 3 colors (27): 27 - 12 = 15.

Therefore, you can make 15 more logos if you use 3 colors instead of 2.

Question 3: How many more logos could you have made using 4 stripes and 2 colors compared to 3 stripes and 2 colors?

To calculate the number of logos you can make with 4 stripes and 2 colors, we use the same logic as before. There are 2 options for each of the first three stripes, and for the fourth stripe, you again have 2 options.

Using the formula, the number of possibilities with 4 stripes and 2 colors would be: 2 * 2 * 2 * 2 = 16.

To calculate the additional logos, subtract the number of logos with 3 stripes and 2 colors (12) from the number of logos with 4 stripes and 2 colors (16): 16 - 12 = 4.

Therefore, you could have made 4 more logos using 4 stripes and 2 colors compared to using 3 stripes and 2 colors.