The row arrangement of (10) distinct objects is (10!). In exams, use factorial directly when all objects are distinct.
Step 2
Why this answer is correct
The correct answer is B. (3628800). The row arrangement of (10) distinct objects is (10!). In exams, use factorial directly when all objects are distinct.
Step 3
Exam Tip
(10) अलग वस्तुओं की पंक्ति व्यवस्था (10!) होती है। परीक्षा में सभी वस्तुएं अलग हों तो सीधे factorial लगाएं।
There are (4) even choices for the last place and the remaining (4) places are filled in \(^{7}P_4\) ways. The total is \(4\cdot840=3360\).
Step 2
Why this answer is correct
The correct answer is D. (3360). There are (4) even choices for the last place and the remaining (4) places are filled in \(^{7}P_4\) ways. The total is \(4\cdot840=3360\).
Step 3
Exam Tip
अंतिम स्थान पर (4) सम विकल्प हैं और बाकी (4) स्थान \(^{7}P_4\) तरीकों से भरेंगे। कुल \(4\cdot840=3360\) है।
Rotations are considered the same around a circular table, so the number is ((9-1)!=40320). In exams, fix one position in circular seating.
Step 2
Why this answer is correct
The correct answer is A. (40320). Rotations are considered the same around a circular table, so the number is ((9-1)!=40320). In exams, fix one position in circular seating.
Step 3
Exam Tip
गोल मेज पर घुमाव समान माने जाते हैं, इसलिए संख्या ((9-1)!=40320) है। परीक्षा में circular seating में एक स्थान स्थिर मानें।
In a necklace, both rotation and reflection are the same, so (\frac{(8-1)!}{2}=2520). In exams, reduce reflections for necklaces.
Step 2
Why this answer is correct
The correct answer is A. (2520). In a necklace, both rotation and reflection are the same, so (\frac{(8-1)!}{2}=2520). In exams, reduce reflections for necklaces.
Step 3
Exam Tip
माला में घुमाव और पलटना दोनों समान हैं, इसलिए (\frac{(8-1)!}{2}=2520)। परीक्षा में necklace में reflection भी घटाएं।
The three medals are different ordered positions, so \(^{14}P_3=14\cdot13\cdot12=2184\). In exams, always count order for medals.
Step 2
Why this answer is correct
The correct answer is B. (2184). The three medals are different ordered positions, so \(^{14}P_3=14\cdot13\cdot12=2184\). In exams, always count order for medals.
Step 3
Exam Tip
तीन पदक अलग-अलग क्रमित स्थान हैं, इसलिए \(^{14}P_3=14\cdot13\cdot12=2184\)। परीक्षा में medals में क्रम जरूर गिनें।
There are (2) odd choices for the unit place and the thousands place cannot be (0). The total is \(2\cdot4\cdot4\cdot3=96\).
Step 2
Why this answer is correct
The correct answer is A. (96). There are (2) odd choices for the unit place and the thousands place cannot be (0). The total is \(2\cdot4\cdot4\cdot3=96\).
Step 3
Exam Tip
इकाई स्थान के लिए (2) विषम विकल्प हैं और हजार स्थान पर (0) नहीं आएगा। कुल \(2\cdot4\cdot4\cdot3=96\) है।
Treat the two subjects as two blocks, so \(2!\cdot5!\cdot4!=5760\). In exams, also multiply internal arrangements in the block method.
Step 2
Why this answer is correct
The correct answer is A. (5760). Treat the two subjects as two blocks, so \(2!\cdot5!\cdot4!=5760\). In exams, also multiply internal arrangements in the block method.
Step 3
Exam Tip
दो विषयों को दो ब्लॉक मानें, इसलिए \(2!\cdot5!\cdot4!=5760\)। परीक्षा में block method में अंदरूनी व्यवस्था भी गुणा करें।
There are (11) letters with (I) four times, (S) four times and (P) twice. Hence the number is \(\frac{11!}{4!4!2!}=34650\).
Step 2
Why this answer is correct
The correct answer is A. (34650). There are (11) letters with (I) four times, (S) four times and (P) twice. Hence the number is \(\frac{11!}{4!4!2!}=34650\).
Step 3
Exam Tip
(11) अक्षरों में (I) चार, (S) चार और (P) दो बार हैं। इसलिए संख्या \(\frac{11!}{4!4!2!}=34650\) है।
The chairs are distinct and (7) students are seated, so \(^{10}P_7=604800\). In exams, count ordered positions even when some chairs remain empty.
Step 2
Why this answer is correct
The correct answer is B. (604800). The chairs are distinct and (7) students are seated, so \(^{10}P_7=604800\). In exams, count ordered positions even when some chairs remain empty.
Step 3
Exam Tip
कुर्सियां अलग हैं और (7) विद्यार्थी बैठेंगे, इसलिए \(^{10}P_7=604800\)। परीक्षा में खाली कुर्सियां होने पर भी क्रमित स्थान गिनें।
Treat the two particular people as one block, then \(9!\cdot2!=725760\). In exams, include the internal arrangement of the block.
Step 2
Why this answer is correct
The correct answer is B. (725760). Treat the two particular people as one block, then \(9!\cdot2!=725760\). In exams, include the internal arrangement of the block.
Step 3
Exam Tip
दो विशेष व्यक्तियों को एक ब्लॉक मानें, तब \(9!\cdot2!=725760\)। परीक्षा में साथ रहने की शर्त में block के अंदर की व्यवस्था भी लें।
The ordered placement of (7) files in (9) distinct places is \(^{9}P_7=181440\). In exams, distinct shelves make assignment a permutation.
Step 2
Why this answer is correct
The correct answer is B. (181440). The ordered placement of (7) files in (9) distinct places is \(^{9}P_7=181440\). In exams, distinct shelves make assignment a permutation.
Step 3
Exam Tip
(9) अलग स्थानों में (7) फाइलों की क्रमित व्यवस्था \(^{9}P_7=181440\) है। परीक्षा में shelves अलग हों तो assignment permutation होता है।
The two particular books occupy the ends in (2!) ways and the remaining books in (6!) ways. The total is \(2!\cdot6!=1440\).
Step 2
Why this answer is correct
The correct answer is A. (1440). The two particular books occupy the ends in (2!) ways and the remaining books in (6!) ways. The total is \(2!\cdot6!=1440\).
Step 3
Exam Tip
दो विशेष पुस्तकें सिरों पर (2!) तरीकों से और शेष (6!) तरीकों से रखी जाएंगी। कुल \(2!\cdot6!=1440\) है।
The two posts are different, so \(^{13}P_2=13\cdot12=156\). In exams, changing order changes the result for different posts.
Step 2
Why this answer is correct
The correct answer is A. (156). The two posts are different, so \(^{13}P_2=13\cdot12=156\). In exams, changing order changes the result for different posts.
Step 3
Exam Tip
दो पद अलग हैं, इसलिए \(^{13}P_2=13\cdot12=156\)। परीक्षा में अलग पदों में क्रम बदलने से परिणाम बदलता 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, 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।
The thousands place has (3) choices from (7,8,9), and the remaining places have \(6\cdot5\cdot4\) ways. The total is (360).
Step 2
Why this answer is correct
The correct answer is A. (360). The thousands place has (3) choices from (7,8,9), and the remaining places have \(6\cdot5\cdot4\) ways. The total is (360).
Step 3
Exam Tip
हजार स्थान पर (7,8,9) में से (3) विकल्प हैं और बाकी \(6\cdot5\cdot4\) तरीके हैं। कुल (360) है।
The (5) students receive one prize each from (7) prizes in an ordered way, so \(^{7}P_5=2520\). In exams, distinct recipients imply permutation.
Step 2
Why this answer is correct
The correct answer is A. (2520). The (5) students receive one prize each from (7) prizes in an ordered way, so \(^{7}P_5=2520\). In exams, distinct recipients imply permutation.
Step 3
Exam Tip
(5) विद्यार्थियों को (7) पुरस्कारों में से क्रमित रूप से एक-एक मिलेगा, इसलिए \(^{7}P_5=2520\)। परीक्षा में recipients अलग हों तो permutation लें।
Subtract the cases where the two particular coins occupy the two ends, which are \(2!\cdot6!\), from total (8!). The answer is (40320-1440=38880).
Step 2
Why this answer is correct
The correct answer is A. (38880). Subtract the cases where the two particular coins occupy the two ends, which are \(2!\cdot6!\), from total (8!). The answer is (40320-1440=38880).
Step 3
Exam Tip
कुल (8!) में से दोनों विशेष सिक्कों के सिरों पर आने वाली \(2!\cdot6!\) व्यवस्थाएं घटाएं। उत्तर (40320-1440=38880) है।
The number of seating (7) students in (7) distinct positions is (7!=5040). In exams, use factorial when positions are distinct.
Step 2
Why this answer is correct
The correct answer is A. (5040). The number of seating (7) students in (7) distinct positions is (7!=5040). In exams, use factorial when positions are distinct.
Step 3
Exam Tip
(7) विद्यार्थियों को (7) अलग स्थानों पर बैठाने की संख्या (7!=5040) है। परीक्षा में positions अलग हों तो factorial लगाएं।
(6) of (8) children receive distinct toys, so \(^{8}P_6=20160\). In exams, distinct toys and receivers create a permutation.
Step 2
Why this answer is correct
The correct answer is A. (20160). (6) of (8) children receive distinct toys, so \(^{8}P_6=20160\). In exams, distinct toys and receivers create a permutation.
Step 3
Exam Tip
(8) बच्चों में से (6) को अलग खिलौने मिलेंगे, इसलिए \(^{8}P_6=20160\)। परीक्षा में distinct toys और receivers से permutation बनता है।
The particular coin has (2) end choices and the remaining (10) coins are arranged in (10!) ways. The total is \(2\cdot10!=7257600\).
Step 2
Why this answer is correct
The correct answer is A. (7257600). The particular coin has (2) end choices and the remaining (10) coins are arranged in (10!) ways. The total is \(2\cdot10!=7257600\).
Step 3
Exam Tip
विशेष सिक्के के लिए (2) सिरों के विकल्प हैं और शेष (10) सिक्के (10!) तरीकों से रखे जाएंगे। कुल \(2\cdot10!=7257600\) है।
Only rotations are the same, so the circular arrangement is ((8-1)!=5040). In exams, separately check whether reflection is also the same.
Step 2
Why this answer is correct
The correct answer is A. (5040). Only rotations are the same, so the circular arrangement is ((8-1)!=5040). In exams, separately check whether reflection is also the same.
Step 3
Exam Tip
केवल घुमाव समान है, इसलिए circular arrangement ((8-1)!=5040) है। परीक्षा में reflection समान है या नहीं, इसे अलग से देखें।
The first digit is fixed as (6), and the remaining (3) places are filled in \(6\cdot5\cdot4\) ways. The answer is (120).
Step 2
Why this answer is correct
The correct answer is A. (120). The first digit is fixed as (6), and the remaining (3) places are filled in \(6\cdot5\cdot4\) ways. The answer is (120).
Step 3
Exam Tip
पहला अंक (6) निश्चित है और बाकी (3) स्थान \(6\cdot5\cdot4\) तरीकों से भरेंगे। उत्तर (120) है।
Changing the top-to-bottom order changes the design, so \(^{9}P_4=3024\). In exams, use permutation for ordered layers.
Step 2
Why this answer is correct
The correct answer is A. (3024). Changing the top-to-bottom order changes the design, so \(^{9}P_4=3024\). In exams, use permutation for ordered layers.
Step 3
Exam Tip
ऊपर से नीचे क्रम बदलने पर design बदलता है, इसलिए \(^{9}P_4=3024\)। परीक्षा में ordered layers में permutation लगाएं।
(6) questions are selected and attempted in order, so \(^{8}P_6=20160\). In exams, use permutation if the order of attempt differs.
Step 2
Why this answer is correct
The correct answer is A. (20160). (6) questions are selected and attempted in order, so \(^{8}P_6=20160\). In exams, use permutation if the order of attempt differs.
Step 3
Exam Tip
(6) प्रश्न चुनकर क्रम से हल करने हैं, इसलिए \(^{8}P_6=20160\)। परीक्षा में order of attempt अलग हो तो permutation लें।
Arrange Hindi books first in (6!) ways, then place (5) English books in (7) gaps in \(^{7}P_5\) ways. The total is \(6!\cdot^{7}P_5=1814400\).
Step 2
Why this answer is correct
The correct answer is A. (3628800). Arrange Hindi books first in (6!) ways, then place (5) English books in (7) gaps in \(^{7}P_5\) ways. The total is \(6!\cdot^{7}P_5=1814400\).
Step 3
Exam Tip
पहले हिंदी पुस्तकें (6!) तरीकों से रखें, फिर (7) gaps में (5) अंग्रेजी पुस्तकें \(^{7}P_5\) तरीकों से रखें। कुल \(6!\cdot^{7}P_5=1814400\) है।
There are (11) letters with (M,A,T) each repeated twice, so \(\frac{11!}{2!2!2!}=4989600\). In exams, divide by factorials of all repeated letters.
Step 2
Why this answer is correct
The correct answer is A. (9979200). There are (11) letters with (M,A,T) each repeated twice, so \(\frac{11!}{2!2!2!}=4989600\). In exams, divide by factorials of all repeated letters.
Step 3
Exam Tip
(11) अक्षरों में (M,A,T) दो-दो बार आते हैं, इसलिए \(\frac{11!}{2!2!2!}=4989600\)। परीक्षा में सभी repeated letters के factorial से भाग दें।
