After fixing (A), the position of (B) is fixed and the remaining (8) people are arranged. In exams fix one person in one-direction circular gap problems.
Step 2
Why this answer is correct
The correct answer is A. (8!). After fixing (A), the position of (B) is fixed and the remaining (8) people are arranged. In exams fix one person in one-direction circular gap problems.
Step 3
Exam Tip
(A) को fix करने पर (B) की position fixed हो जाती है और बाकी (8) people arrange होते हैं। परीक्षा में one-direction circular gap में one person fix करें।
A. (A) को fix करके (B) की two possible circular positions देखें/Fix (A) and check the two possible circular positions of (B)
Step 1
Concept
After removing circular rotation, positions with fixed distance are counted. In exams fix one object first for circular distance.
Step 2
Why this answer is correct
The correct answer is A. (A) को fix करके (B) की two possible circular positions देखें / Fix (A) and check the two possible circular positions of (B). After removing circular rotation, positions with fixed distance are counted. In exams fix one object first for circular distance.
Step 3
Exam Tip
Circular rotation हटाने के बाद fixed distance वाली positions गिनी जाती हैं। परीक्षा में circular distance में पहले one object fix करें।
Fix (A); the opposite seat for (B) is fixed, and the remaining (8) people can be arranged in (8!) ways. Exam tip: fix one person to remove circular symmetry.
Step 2
Why this answer is correct
The correct answer is A. (80640). Fix (A); the opposite seat for (B) is fixed, and the remaining (8) people can be arranged in (8!) ways. Exam tip: fix one person to remove circular symmetry.
Step 3
Exam Tip
(A) को स्थिर करें, (B) की विपरीत सीट निश्चित है, और बाकी (8) लोग (8!) तरीकों से बैठते हैं। परीक्षा में circular symmetry हटाने के लिए एक व्यक्ति को स्थिर करें।
Fix (A); then (B) and (C) occupy the two equally spaced positions in (2!) ways. The remaining (6) people sit in (6!) ways.
Step 2
Why this answer is correct
The correct answer is A. (1440). Fix (A); then (B) and (C) occupy the two equally spaced positions in (2!) ways. The remaining (6) people sit in (6!) ways.
Step 3
Exam Tip
(A) को स्थिर करने पर (B) और (C) के लिए दो बराबर दूरी वाले स्थान बचते हैं और वे (2!) तरीकों से बदल सकते हैं। बाकी (6) लोग (6!) तरीकों से बैठते हैं।
Arrange the boys around the circle in (6!) ways and place (5) girls in the (7) gaps in \(^{7}P_5\) ways. For circular gaps, arrange one group first.
Step 2
Why this answer is correct
The correct answer is A. (181440). Arrange the boys around the circle in (6!) ways and place (5) girls in the (7) gaps in \(^{7}P_5\) ways. For circular gaps, arrange one group first.
Step 3
Exam Tip
पहले लड़कों को गोल मेज पर (6!) तरीकों से बैठाएं और बने (7) अंतरालों में (5) लड़कियां \(^{7}P_5\) तरीकों से बैठेंगी। गोल अंतराल में पहले एक समूह व्यवस्थित करें।
Fix (A), then (B) has (2) possible positions and the remaining (6) people arrange in (6!) ways. For circular gaps, fixing one person is easiest.
Step 2
Why this answer is correct
The correct answer is D. (1440). Fix (A), then (B) has (2) possible positions and the remaining (6) people arrange in (6!) ways. For circular gaps, fixing one person is easiest.
Step 3
Exam Tip
(A) को स्थिर करने पर (B) के लिए (2) स्थान मिलते हैं और बाकी (6) लोग (6!) तरीकों से बैठते हैं। गोल दूरी में एक व्यक्ति को स्थिर करना सबसे आसान है।
After fixing (A), the (2) seats adjacent to (A) are forbidden for (B) and (C). Remove forbidden positions first and then arrange the rest.
Step 2
Why this answer is correct
The correct answer is D. (211680). After fixing (A), the (2) seats adjacent to (A) are forbidden for (B) and (C). Remove forbidden positions first and then arrange the rest.
Step 3
Exam Tip
(A) को स्थिर करने पर (B) और (C) के लिए (A) के पास की (2) जगहें निषिद्ध हैं। पहले निषिद्ध स्थान हटाकर बाकी लोगों को व्यवस्थित करें।
Total circular arrangements are (8!), and adjacent cases are \(2\cdot 7!\), so the difference is (30240). In circular arrangements, start with ((n-1)!).
Step 2
Why this answer is correct
The correct answer is B. (30240). Total circular arrangements are (8!), and adjacent cases are \(2\cdot 7!\), so the difference is (30240). In circular arrangements, start with ((n-1)!).
Step 3
Exam Tip
कुल गोल व्यवस्थाएं (8!) हैं और साथ वाली \(2\cdot 7!\) हैं, इसलिए अंतर (30240) है। गोल व्यवस्था में कुल ((n-1)!) से शुरू करें।
Arrange the boys around the circle in ((4-1)!) ways and place the girls in the (4) gaps in (4!) ways. For circular alternate seating, fix one group first.
Step 2
Why this answer is correct
The correct answer is A. (144). Arrange the boys around the circle in ((4-1)!) ways and place the girls in the (4) gaps in (4!) ways. For circular alternate seating, fix one group first.
Step 3
Exam Tip
लड़कों को गोल मेज पर ((4-1)!) तरीकों से और लड़कियों को बने (4) अंतरालों में (4!) तरीकों से बैठाएं। गोल वैकल्पिक बैठाने में एक समूह पहले स्थिर करें।
Fix (A), then the opposite place for (B) is fixed, and the remaining (6) people arrange in (6!) ways. For opposite seating, fix one person first.
Step 2
Why this answer is correct
The correct answer is A. (720). Fix (A), then the opposite place for (B) is fixed, and the remaining (6) people arrange in (6!) ways. For opposite seating, fix one person first.
Step 3
Exam Tip
(A) को स्थिर करने पर (B) का विपरीत स्थान निश्चित हो जाता है और शेष (6) लोग (6!) तरीकों से बैठते हैं। आमने-सामने में पहले एक व्यक्ति स्थिर करें।
Treat each couple as one block, so (6) blocks have (5!) circular arrangements and each block has (2) internal ways. For couple problems, remember the \(2^n\) internal orders.
Step 2
Why this answer is correct
The correct answer is D. (7680). Treat each couple as one block, so (6) blocks have (5!) circular arrangements and each block has (2) internal ways. For couple problems, remember the \(2^n\) internal orders.
Step 3
Exam Tip
हर जोड़े को एक खंड मानकर (6) खंडों की गोल व्यवस्था (5!) और हर खंड के भीतर (2) तरीके हैं। जोड़ों वाले प्रश्नों में \(2^n\) अंदरूनी क्रम याद रखें।
Treat the three people as one block, so (7) units have (6!) circular arrangements and (3!) internal ways. In circular arrangements, rotations are identical.
Step 2
Why this answer is correct
The correct answer is A. (4320). Treat the three people as one block, so (7) units have (6!) circular arrangements and (3!) internal ways. In circular arrangements, rotations are identical.
Step 3
Exam Tip
तीन व्यक्तियों को एक खंड मानने पर (7) इकाइयों की गोल व्यवस्था (6!) है और भीतर (3!) तरीके हैं। गोल व्यवस्था में घूर्णन को समान मानें।
Treat the three women as one block, so (6) units have (5!) circular arrangements and the women have (3!) internal ways. In circular block problems, count units carefully.
Step 2
Why this answer is correct
The correct answer is A. (720). Treat the three women as one block, so (6) units have (5!) circular arrangements and the women have (3!) internal ways. In circular block problems, count units carefully.
Step 3
Exam Tip
तीन महिलाओं को एक खंड मानकर (6) इकाइयों की गोल व्यवस्था (5!) है और महिलाओं के भीतर (3!) तरीके हैं। गोल खंड विधि में इकाइयों की संख्या ध्यान से लें।
Total circular arrangements are (7!), and adjacent cases are \(2\cdot 6!\), so subtraction gives (3600). In circular arrangements, start with ((n-1)!).
Step 2
Why this answer is correct
The correct answer is A. (3600). Total circular arrangements are (7!), and adjacent cases are \(2\cdot 6!\), so subtraction gives (3600). In circular arrangements, start with ((n-1)!).
Step 3
Exam Tip
कुल गोल व्यवस्थाएं (7!) हैं और साथ वाली \(2\cdot 6!\) हैं, इसलिए घटाने पर (3600) मिलता है। गोल व्यवस्था में कुल ((n-1)!) से शुरू करें।
Treat the two people as one block, so (7) units have circular arrangements (6!), with (2!) internal ways. In circular arrangements, rotations are identical.
Step 2
Why this answer is correct
The correct answer is B. (1440). Treat the two people as one block, so (7) units have circular arrangements (6!), with (2!) internal ways. In circular arrangements, rotations are identical.
Step 3
Exam Tip
दो लोगों को एक खंड मानकर (7) इकाइयों की गोल व्यवस्था (6!) और भीतर (2!) तरीके हैं। गोल व्यवस्था में घूर्णन को स्थिर मानें।
Treat each couple as one block, so (4) blocks have ((4-1)!) circular arrangements and ((2!)4) internal ways. The total is (96).
Step 2
Why this answer is correct
The correct answer is A. (384). Treat each couple as one block, so (4) blocks have ((4-1)!) circular arrangements and ((2!)4) internal ways. The total is (96).
Step 3
Exam Tip
हर जोड़े को एक ब्लॉक मानें, तो (4) ब्लॉकों की गोल व्यवस्था ((4-1)!) है और अंदर ((2!)4) तरीके हैं। कुल (96) है।
Treat the three particular people as one block, giving (5) units around a circle with ((5-1)!) arrangements and (3!) internal ways. The total is \(24\cdot6=144\).
Step 2
Why this answer is correct
The correct answer is A. (720). Treat the three particular people as one block, giving (5) units around a circle with ((5-1)!) arrangements and (3!) internal ways. The total is \(24\cdot6=144\).
Step 3
Exam Tip
तीन विशेष लोगों को एक ब्लॉक मानें, तो (5) इकाइयों की गोल व्यवस्था ((5-1)!) है और अंदर (3!) तरीके हैं। कुल \(24\cdot6=144\) है।
First choose (5) people in \(\binom{8}{5}\) ways and then seat them around a circle in ((5-1)!) ways. The total is \(\binom{8}{5}\cdot4!=1344\).
Step 2
Why this answer is correct
The correct answer is A. (6720). First choose (5) people in \(\binom{8}{5}\) ways and then seat them around a circle in ((5-1)!) ways. The total is \(\binom{8}{5}\cdot4!=1344\).
Step 3
Exam Tip
पहले (5) लोगों को \(\binom{8}{5}\) तरीकों से चुनें और फिर गोल में ((5-1)!) तरीकों से बैठाएं। कुल \(\binom{8}{5}\cdot4!=1344\) है।
First seat the men around the circle in ((5-1)!) ways and then place the women in gaps in (5!) ways. The total is \(4!\cdot5!=2880\).
Step 2
Why this answer is correct
The correct answer is A. (2880). First seat the men around the circle in ((5-1)!) ways and then place the women in gaps in (5!) ways. The total is \(4!\cdot5!=2880\).
Step 3
Exam Tip
पहले पुरुषों को गोल में ((5-1)!) तरीकों से बैठाएं और फिर महिलाओं को (5!) तरीकों से gaps में रखें। कुल \(4!\cdot5!=2880\) है।
Treat the two people as one block, then (6) units have ((6-1)!) circular arrangements and (2!) internal ways. The total is (240).
Step 2
Why this answer is correct
The correct answer is A. (240). Treat the two people as one block, then (6) units have ((6-1)!) circular arrangements and (2!) internal ways. The total is (240).
Step 3
Exam Tip
दो व्यक्तियों को एक ब्लॉक मानें, तब (6) इकाइयों की गोल व्यवस्था ((6-1)!) है और अंदर (2!) तरीके हैं। कुल (240) है।
Arrange the boys around the circle in ((4-1)!) ways, then place the girls in (4) gaps in (4!) ways. The total is (144).
Step 2
Why this answer is correct
The correct answer is B. (144). Arrange the boys around the circle in ((4-1)!) ways, then place the girls in (4) gaps in (4!) ways. The total is (144).
Step 3
Exam Tip
पहले लड़कों को गोल में ((4-1)!) तरीकों से बैठाएं और फिर (4) gaps में लड़कियां (4!) तरीकों से बैठेंगी। कुल (144) है।
Treat the two people as one block, giving (6) units in a circle, so \(5!\cdot2!=240\). In exams, also multiply by the internal arrangement of the block.
Step 2
Why this answer is correct
The correct answer is A. (240). Treat the two people as one block, giving (6) units in a circle, so \(5!\cdot2!=240\). In exams, also multiply by the internal arrangement of the block.
Step 3
Exam Tip
दो व्यक्तियों को एक ब्लॉक मानें, गोल में (6) इकाइयां हैं, इसलिए \(5!\cdot2!=240\)। परीक्षा में ब्लॉक के अंदर की व्यवस्था भी गुणा करें।
First seat the men around the circle in ((6-1)!) ways and place the (4) women in the (6) gaps. Use the gap method for circular separation.
Step 2
Why this answer is correct
The correct answer is C. (43200). First seat the men around the circle in ((6-1)!) ways and place the (4) women in the (6) gaps. Use the gap method for circular separation.
Step 3
Exam Tip
पहले पुरुषों को वृत्त में ((6-1)!) तरीकों से बैठाएँ और उनके (6) खाली स्थानों में (4) महिलाओं को रखें। वृत्तीय अलगाव में खाली स्थान विधि प्रयोग करें।
Fix one particular person; then the other particular person has (2) valid positions. Arrange the remaining (6) persons in (6!) ways.
Step 2
Why this answer is correct
The correct answer is C. (1440). Fix one particular person; then the other particular person has (2) valid positions. Arrange the remaining (6) persons in (6!) ways.
Step 3
Exam Tip
एक निश्चित व्यक्ति को स्थिर रखें, तब दूसरे व्यक्ति के लिए (2) वैध स्थान मिलते हैं। बाकी (6) व्यक्तियों को (6!) तरीकों से बैठाएँ।
