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Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in how many different ways can the women pose?
- a)2
- b)14
- c)15
- d)16
- e)18
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Eight women of eight different heights are to pose for a photo in two ...
This is a counting problem that is best solved using logic. First, let’s represent the line of women as follows:
0000
0000
where the heights go from 1 to 8 in increasing order and the unknowns are designated 0s. Since the women are arranged by their heights in increasing order from left to right and front to back, we know that at a minimum, the lineup must conform to this:
0008
1000
Let’s further designate the arrangement by labeling the other individuals in the top row as X, Y and Z, and the individuals in the bottom row as A, B, and C.
XYZ8
1ABC
Note that Z must be greater than at least 5 numbers (X, Y, B, A, and 1) and less than at least 1 number (8). This means that Z can only be a 6 or a 7. Note that Y must be greater than at least 3 numbers (X, A and 1) and less than at least 2 numbers (8 and Z). This means that Y can only be 4, 5, or 6. Note that X must be greater than at least 1 number (1) and less than at least 3 numbers (8, Z and Y), This means that X must be 2, 3, 4, or 5. This is enough information to start counting the total number of possibilities for the top row. It will be easiest to use the middle unknown value Y as our starting point. As we determined above, Y can only be 4, 5, or 6. Let’s check each case, making our conclusions logically: If Y is 4, Z has 2 options (6 or 7) and X has 2 options (2 or 3). This yields 2 x 2 = 4 possibilities. If Y is 5, Z has 2 options (6 or 7), and X has 3 options (2, 3, or 4). This yields 2 x 3 = 6 possibilities. If Y is 6, Z has 1 option (7), and X has 4 options (2, 3, 4, or 5). This yields 1 x 4 = 4 possibilities. For each of the possibilities above, the bottom row is completely determined because we have 3 numbers left, all of
which must be in placed increasing order. Hence, there are 4 + 6 + 4 = 14 ways for the women to pose.
0000
0000
where the heights go from 1 to 8 in increasing order and the unknowns are designated 0s. Since the women are arranged by their heights in increasing order from left to right and front to back, we know that at a minimum, the lineup must conform to this:
0008
1000
Let’s further designate the arrangement by labeling the other individuals in the top row as X, Y and Z, and the individuals in the bottom row as A, B, and C.
XYZ8
1ABC
Note that Z must be greater than at least 5 numbers (X, Y, B, A, and 1) and less than at least 1 number (8). This means that Z can only be a 6 or a 7. Note that Y must be greater than at least 3 numbers (X, A and 1) and less than at least 2 numbers (8 and Z). This means that Y can only be 4, 5, or 6. Note that X must be greater than at least 1 number (1) and less than at least 3 numbers (8, Z and Y), This means that X must be 2, 3, 4, or 5. This is enough information to start counting the total number of possibilities for the top row. It will be easiest to use the middle unknown value Y as our starting point. As we determined above, Y can only be 4, 5, or 6. Let’s check each case, making our conclusions logically: If Y is 4, Z has 2 options (6 or 7) and X has 2 options (2 or 3). This yields 2 x 2 = 4 possibilities. If Y is 5, Z has 2 options (6 or 7), and X has 3 options (2, 3, or 4). This yields 2 x 3 = 6 possibilities. If Y is 6, Z has 1 option (7), and X has 4 options (2, 3, 4, or 5). This yields 1 x 4 = 4 possibilities. For each of the possibilities above, the bottom row is completely determined because we have 3 numbers left, all of
which must be in placed increasing order. Hence, there are 4 + 6 + 4 = 14 ways for the women to pose.
Most Upvoted Answer
Eight women of eight different heights are to pose for a photo in two ...
To solve this problem, we need to consider the different arrangements of the women in the two rows while adhering to the given restrictions. Let's break down the solution step by step.
1. Determine the number of ways to arrange the women in the first row:
- Since there are eight women of different heights, we need to select four of them to be in the first row.
- The number of ways to choose four women out of eight is given by the combination formula: C(8, 4) = 8! / (4! * (8-4)!) = 70.
2. Determine the number of ways to arrange the women in the second row:
- For each arrangement in the first row, we need to determine the possible arrangements in the second row.
- The women in the second row must stand directly behind a shorter woman in the first row. This means that the tallest woman in the second row must be placed behind the tallest woman in the first row, the second tallest woman in the second row behind the second tallest woman in the first row, and so on.
- Once the tallest woman in the second row is placed, there are three remaining spots for the other three women in the second row.
- The number of ways to arrange these three women is given by the permutation formula: P(4, 3) = 4! / (4-3)! = 4.
3. Calculate the total number of ways to pose:
- To find the total number of ways, we need to multiply the number of ways to arrange the women in the first row by the number of ways to arrange the women in the second row for each arrangement in the first row.
- Total number of ways = 70 (from step 1) * 4 (from step 2) = 280.
However, we need to consider that the women in each row must be arranged in order of increasing height from left to right. This means that we need to divide the total number of ways by the number of ways to arrange the women within each row. Since there are four women in each row, the number of ways to arrange them in increasing order is 4!.
Final answer:
The number of different ways the women can pose is 280 / (4!) = 280 / 24 = 14.
Therefore, the correct answer is option (B) 14.
1. Determine the number of ways to arrange the women in the first row:
- Since there are eight women of different heights, we need to select four of them to be in the first row.
- The number of ways to choose four women out of eight is given by the combination formula: C(8, 4) = 8! / (4! * (8-4)!) = 70.
2. Determine the number of ways to arrange the women in the second row:
- For each arrangement in the first row, we need to determine the possible arrangements in the second row.
- The women in the second row must stand directly behind a shorter woman in the first row. This means that the tallest woman in the second row must be placed behind the tallest woman in the first row, the second tallest woman in the second row behind the second tallest woman in the first row, and so on.
- Once the tallest woman in the second row is placed, there are three remaining spots for the other three women in the second row.
- The number of ways to arrange these three women is given by the permutation formula: P(4, 3) = 4! / (4-3)! = 4.
3. Calculate the total number of ways to pose:
- To find the total number of ways, we need to multiply the number of ways to arrange the women in the first row by the number of ways to arrange the women in the second row for each arrangement in the first row.
- Total number of ways = 70 (from step 1) * 4 (from step 2) = 280.
However, we need to consider that the women in each row must be arranged in order of increasing height from left to right. This means that we need to divide the total number of ways by the number of ways to arrange the women within each row. Since there are four women in each row, the number of ways to arrange them in increasing order is 4!.
Final answer:
The number of different ways the women can pose is 280 / (4!) = 280 / 24 = 14.
Therefore, the correct answer is option (B) 14.
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Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in howmany different ways can the women pose?a)2b)14c)15d)16e)18Correct answer is option 'B'. Can you explain this answer?
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Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in howmany different ways can the women pose?a)2b)14c)15d)16e)18Correct answer is option 'B'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in howmany different ways can the women pose?a)2b)14c)15d)16e)18Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in howmany different ways can the women pose?a)2b)14c)15d)16e)18Correct answer is option 'B'. Can you explain this answer?.
Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in howmany different ways can the women pose?a)2b)14c)15d)16e)18Correct answer is option 'B'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in howmany different ways can the women pose?a)2b)14c)15d)16e)18Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in howmany different ways can the women pose?a)2b)14c)15d)16e)18Correct answer is option 'B'. Can you explain this answer?.
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