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A new flag is to be designed with six vertical stripes using some or all of the colours- yellow, green, blue and red. Then the number of ways this can be done such that no two adjacent stripes have the same colour out is:
- a)12×81
- b)16×25
- c)20×125
- d)24×216
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A new flag is to be designed with six vertical stripes using some or a...
Since no adjacent stripes contain the same color. The second, and the remaining vertical
stripe will have only three options, while the first will have any of the four options.
4*3*3*3*3*3 = 12* 3^4 = 12 * 81.
So the answer is A.
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A new flag is to be designed with six vertical stripes using some or a...
Explanation:
To design a flag with six vertical stripes using the colors yellow, green, blue, and red, we need to ensure that no two adjacent stripes have the same color. We can approach this problem using a recursive method.
Step 1: Determine the number of ways to arrange the first two stripes.
- We have four colors to choose from for the first stripe.
- For the second stripe, we have three colors available that are different from the color chosen for the first stripe.
- Therefore, the number of ways to arrange the first two stripes is 4 × 3 = 12.
Step 2: Determine the number of ways to arrange the remaining four stripes.
- After arranging the first two stripes, we are left with four stripes to arrange.
- For each of the remaining four stripes, we have three colors available that are different from the color of the stripe immediately preceding it.
- Therefore, the number of ways to arrange the remaining four stripes is 3 × 3 × 3 × 3 = 81.
Step 3: Calculate the total number of ways to arrange all six stripes.
- The total number of ways to arrange all six stripes is obtained by multiplying the number of ways to arrange the first two stripes with the number of ways to arrange the remaining four stripes.
- Therefore, the total number of ways is 12 × 81 = 972.
However, we need to consider that the order of the stripes is not important. For example, the flag with stripes arranged as yellow-green-blue-red-yellow-green is considered the same as the flag with stripes arranged as green-blue-red-yellow-green-yellow. Therefore, we need to divide the total number of arrangements by the number of possible permutations of six stripes, which is 6!.
Step 4: Calculate the final answer.
- The final answer is obtained by dividing the total number of arrangements by the number of possible permutations.
- 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
- Therefore, the final answer is 972 / 720 = 1.35 (approximately).
Since the number of ways to arrange the stripes must be a whole number, we need to find the closest whole number to 1.35.
1.35 is closest to 1, so the correct answer is 12 × 81 = 972, which is option A.
To design a flag with six vertical stripes using the colors yellow, green, blue, and red, we need to ensure that no two adjacent stripes have the same color. We can approach this problem using a recursive method.
Step 1: Determine the number of ways to arrange the first two stripes.
- We have four colors to choose from for the first stripe.
- For the second stripe, we have three colors available that are different from the color chosen for the first stripe.
- Therefore, the number of ways to arrange the first two stripes is 4 × 3 = 12.
Step 2: Determine the number of ways to arrange the remaining four stripes.
- After arranging the first two stripes, we are left with four stripes to arrange.
- For each of the remaining four stripes, we have three colors available that are different from the color of the stripe immediately preceding it.
- Therefore, the number of ways to arrange the remaining four stripes is 3 × 3 × 3 × 3 = 81.
Step 3: Calculate the total number of ways to arrange all six stripes.
- The total number of ways to arrange all six stripes is obtained by multiplying the number of ways to arrange the first two stripes with the number of ways to arrange the remaining four stripes.
- Therefore, the total number of ways is 12 × 81 = 972.
However, we need to consider that the order of the stripes is not important. For example, the flag with stripes arranged as yellow-green-blue-red-yellow-green is considered the same as the flag with stripes arranged as green-blue-red-yellow-green-yellow. Therefore, we need to divide the total number of arrangements by the number of possible permutations of six stripes, which is 6!.
Step 4: Calculate the final answer.
- The final answer is obtained by dividing the total number of arrangements by the number of possible permutations.
- 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
- Therefore, the final answer is 972 / 720 = 1.35 (approximately).
Since the number of ways to arrange the stripes must be a whole number, we need to find the closest whole number to 1.35.
1.35 is closest to 1, so the correct answer is 12 × 81 = 972, which is option A.
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A new flag is to be designed with six vertical stripes using some or all of the colours- yellow, green, blue and red. Then the number of ways this can be done such that no two adjacent stripes have the same colour out is:a)12×81b)16×25c) 20×125d) 24×216Correct answer is option 'A'. Can you explain this answer?
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