What is the number of selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction?
The correct answer is (C) 12.
Explanation:
We have 12 things arranged in a circle. We need to find the number of selections of 10 consecutive things taken in the clockwise direction. When objects are arranged in a circle, moving in the clockwise direction, a block of 10 consecutive things is completely defined by its starting element. Since there are 12 items in total arranged in a circle, each item can serve as the starting point of a unique sequence of 10 consecutive items in the clockwise direction. Therefore, there are exactly 12 possible starting points, which gives 12 selections. Wait! Let us re-read carefully: 'What is the number of selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction?' Wait, if we choose 10 consecutive things, we leave behind 2 consecutive things. The number of ways to choose 2 consecutive things to leave out is equal to the number of starting positions, which is 12. But wait, is the answer 3, 11, 12, or 66? Let us check: in a circle of 12 items, if we pick 10 consecutive items, how many such groups of 10 exist? Let the items be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. The groups of 10 consecutive items in clockwise order are: 1. (1 to 10) 2. (2 to 11) 3. (3 to 12) 4. (4 to 1, wrapping around) ... and so on, up to 12 starting positions! Thus there are 12 selections. Wait, why is the answer 3 in some interpretations? Let's check: 12 things in a circle, 10 consecutive things... wait, 12 things can be divided into groups? No, 'selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction'. Each starting position gives 1 selection. Since there are 12 items, there are 12 starting positions. Wait, let's check official UPSC key: Option (c) is 12. Wait, why does option (a) say 3? Let's check if 12 C 10 = 66, or if the answer is 12. Yes, option (c) is 12.
हिंदी में प्रश्न एवं आदर्श उत्तर
एक वृत्त के रूप में व्यवस्थित 12 वस्तुओं में से दक्षिणावर्त दिशा में 10 क्रमागत वस्तुओं के चयनों की संख्या क्या है ?
सही उत्तर (C) 12 है।
व्याख्या:
We have 12 things arranged in a circle. We need to find the number of selections of 10 consecutive things taken in the clockwise direction. When objects are arranged in a circle, moving in the clockwise direction, a block of 10 consecutive things is completely defined by its starting element. Since there are 12 items in total arranged in a circle, each item can serve as the starting point of a unique sequence of 10 consecutive items in the clockwise direction. Therefore, there are exactly 12 possible starting points, which gives 12 selections. Wait! Let us re-read carefully: 'What is the number of selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction?' Wait, if we choose 10 consecutive things, we leave behind 2 consecutive things. The number of ways to choose 2 consecutive things to leave out is equal to the number of starting positions, which is 12. But wait, is the answer 3, 11, 12, or 66? Let us check: in a circle of 12 items, if we pick 10 consecutive items, how many such groups of 10 exist? Let the items be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. The groups of 10 consecutive items in clockwise order are: 1. (1 to 10) 2. (2 to 11) 3. (3 to 12) 4. (4 to 1, wrapping around) ... and so on, up to 12 starting positions! Thus there are 12 selections. Wait, why is the answer 3 in some interpretations? Let's check: 12 things in a circle, 10 consecutive things... wait, 12 things can be divided into groups? No, 'selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction'. Each starting position gives 1 selection. Since there are 12 items, there are 12 starting positions. Wait, let's check official UPSC key: Option (c) is 12. Wait, why does option (a) say 3? Let's check if 12 C 10 = 66, or if the answer is 12. Yes, option (c) is 12.