Level 58 • 50/50 questions • 35 seconds per question.
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In how many ways can (8) distinct books be arranged in a row?
A. (40320)
अलग-अलग वस्तुओं की रैखिक व्यवस्था (n!) होती है। परीक्षा में पहले वस्तुओं की संख्या पहचानें। / A linear arrangement of distinct objects is (n!). In exams, first identify the number of objects.
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A. (90)
अलग पदों के लिए क्रम महत्वपूर्ण होता है, इसलिए \(^{10}P_2=90\)। परीक्षा में पद अलग हों तो संयोजन नहीं लगाएं। / For different posts, order matters, so \(^{10}P_2=90\). In exams, do not use combinations when posts are distinct.
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A. (60)
यहां (5) अक्षरों में (2) समान हैं, इसलिए संख्या \(\frac{5!}{2!}=60\) है। परीक्षा में समान अक्षरों से भाग देना न भूलें। / There are (5) letters with (2) identical letters, so the number is \(\frac{5!}{2!}=60\). In exams, remember to divide by identical letters.
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In how many ways can first, second and third positions be awarded among (7) distinct players?
A. (210)
तीन क्रमित स्थानों के लिए \(^{7}P_3=7\cdot6\cdot5=210\)। परीक्षा में स्थानों का क्रम हमेशा महत्वपूर्ण मानें। / For three ordered positions, \(^{7}P_3=7\cdot6\cdot5=210\). In exams, ranks always mean order matters.
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In how many ways can (4) students be seated on (6) distinct chairs?
A. (360)
पहले कुर्सियां अलग हैं और (4) विद्यार्थी बैठेंगे, इसलिए \(^{6}P_4=360\)। परीक्षा में खाली कुर्सियों को भी ध्यान में रखें। / The chairs are distinct and (4) students are seated, so \(^{6}P_4=360\). In exams, account for unused chairs carefully.
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How many (3)-digit numbers can be formed from (1,2,3,4,5) without repetition?
A. (60)
तीन स्थानों के लिए (5,4,3) विकल्प हैं, इसलिए \(5\cdot4\cdot3=60\)। परीक्षा में बिना पुनरावृत्ति में विकल्प घटते हैं। / There are (5,4,3) choices for the three places, so \(5\cdot4\cdot3=60\). In exams, choices reduce without repetition.
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How many (3)-digit numbers can be formed from (0,1,2,3,4) without repetition?
A. (48)
सैकड़े के स्थान पर (0) नहीं आ सकता, इसलिए \(4\cdot4\cdot3=48\)। परीक्षा में प्रथम अंक पर शून्य की शर्त जरूर जांचें। / The hundreds place cannot be (0), so \(4\cdot4\cdot3=48\). In exams, always check the zero restriction on the first digit.
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How many different arrangements can be made from the letters of the word BHARAT?
A. (360)
कुल (6) अक्षरों में (2) समान हैं, इसलिए \(\frac{6!}{2!}=360\)। परीक्षा में समान अक्षरों का factorial हर में रखें। / There are (6) letters with (2) identical letters, so \(\frac{6!}{2!}=360\). In exams, put factorials of repeated letters in the denominator.
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In how many orders can a signal of (4) flags be made from (9) distinct flags?
A. (3024)
संकेत में क्रम महत्वपूर्ण है, इसलिए \(^{9}P_4=3024\)। परीक्षा में संकेत और कोड जैसे शब्दों में permutation सोचें। / Order is important in a signal, so \(^{9}P_4=3024\). In exams, words like signal and code usually indicate permutation.
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In how many ways can (5) boys and (4) girls be seated in a row so that all girls sit together?
A. (17280)
चार लड़कियों को एक ब्लॉक मानें, तब \(6!\cdot4!=17280\)। परीक्षा में साथ बैठने की शर्त में ब्लॉक विधि लगाएं। / Treat the four girls as one block, then \(6!\cdot4!=17280\). In exams, use the block method for togetherness conditions.
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In how many ways can (6) people be seated around a circular table?
A. (120)
गोल व्यवस्था में संख्या ((n-1)!) होती है, इसलिए (5!=120)। परीक्षा में घुमाव को समान मानें। / In a circular arrangement, the number is ((n-1)!), so (5!=120). In exams, rotations are considered the same.
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A. (240)
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। / 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.
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How many distinct arrangements are possible using all letters of SAMITI?
A. (360)
(6) अक्षरों में (2) अक्षर समान हैं, इसलिए \(\frac{6!}{2!}=360\)। परीक्षा में केवल समान अक्षरों की पुनरावृत्ति गिनें। / Among (6) letters, (2) are identical, so \(\frac{6!}{2!}=360\). In exams, count only the repeated identical letters.
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How many (4)-digit even numbers can be formed from (2,3,4,5,6,7) without repetition?
A. (180)
इकाई स्थान पर (3) सम विकल्प हैं और बाकी \(5\cdot4\cdot3\) तरीके हैं, इसलिए (180)। परीक्षा में अंतिम अंक की शर्त पहले लगाएं। / There are (3) even choices for the units place and \(5\cdot4\cdot3\) ways for the rest, so (180). In exams, apply the last-digit condition first.
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How many (4)-digit odd numbers can be formed from (1,2,3,4,5,6) without repetition?
A. (180)
इकाई स्थान पर (3) विषम अंक हैं और शेष स्थान \(5\cdot4\cdot3\) तरीकों से भरते हैं। कुल (180) संख्याएं मिलती हैं। / There are (3) odd choices for the units place and the remaining places are filled in \(5\cdot4\cdot3\) ways. The total is (180).
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A. (1440)
दो विषयों को दो ब्लॉक मानें, इसलिए \(2!\cdot5!\cdot3!=1440\)। परीक्षा में हर ब्लॉक की अंदरूनी व्यवस्था अलग से करें। / Treat the two subjects as two blocks, so \(2!\cdot5!\cdot3!=1440\). In exams, arrange inside each block separately.
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How many (5)-letter passwords can be formed from (8) distinct letters if no letter is repeated?
A. (6720)
पासवर्ड में क्रम महत्वपूर्ण है, इसलिए \(^{8}P_5=6720\)। परीक्षा में पासवर्ड को arrangement की तरह लें। / Order matters in a password, so \(^{8}P_5=6720\). In exams, treat passwords as arrangements.
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How many ways can the letters of GANIT be arranged?
A. (120)
सभी (5) अक्षर अलग हैं, इसलिए (5!=120)। परीक्षा में समान अक्षर न हों तो सीधे (n!) लगाएं। / All (5) letters are distinct, so (5!=120). In exams, use (n!) directly when there are no repeated letters.
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What is the number of ways to arrange (3) objects selected from (11) objects in order?
A. (990)
क्रम सहित चयन \(^{11}P_3=11\cdot10\cdot9=990\) है। परीक्षा में arranged शब्द दिखे तो permutation लें। / Selection with order is \(^{11}P_3=11\cdot10\cdot9=990\). In exams, the word arranged suggests permutation.
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A. (576)
चार लाल गेंदों को एक ब्लॉक मानें, तब \(4!\cdot4!=576\)। परीक्षा में ब्लॉक और बाहरी इकाइयों को अलग-अलग व्यवस्थित करें। / Treat the four red balls as one block, then \(4!\cdot4!=576\). In exams, arrange the block and outside units separately.
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How many (3)-digit even numbers can be formed from (0,2,4,6,8) without repetition?
A. (48)
सभी उपलब्ध अंतिम अंक सम हैं, पर सैकड़े में (0) नहीं हो सकता। कुल \(4\cdot4\cdot3=48\) संख्याएं बनेंगी। / All possible last digits are even, but the hundreds place cannot be (0). Total numbers are \(4\cdot4\cdot3=48\).
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In how many ways can (5) distinct persons be placed one each in (5) distinct rooms?
A. (120)
यह (5) व्यक्तियों और (5) कमरों की एक-एक व्यवस्था है, इसलिए (5!=120)। परीक्षा में distinct rooms हों तो assignment भी permutation है। / This is a one-to-one arrangement of (5) persons and (5) rooms, so (5!=120). In exams, assignments to distinct rooms are permutations.
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In how many ways can a captain, vice-captain and treasurer be appointed from (9) students?
A. (504)
तीन अलग पद हैं, इसलिए \(^{9}P_3=504\)। परीक्षा में पदों के नाम अलग हों तो क्रम बदलने से परिणाम बदलता है। / There are three different posts, so \(^{9}P_3=504\). In exams, different named posts make order important.
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In how many ways can (6) distinct keys be placed against (6) distinct locks?
A. (720)
छह अलग वस्तुओं का छह अलग स्थानों पर विन्यास (6!=720) है। परीक्षा में matching positions अलग हों तो factorial लगाएं। / The arrangement of six distinct objects in six distinct positions is (6!=720). In exams, use factorial when matching positions are distinct.
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A. (181440)
(9) अक्षरों में (3) अक्षर समान हैं, इसलिए \(\frac{9!}{3!}=60480\); लेकिन अंग्रेजी शब्द में (A) दो बार और (Y) दो बार हैं, अतः \(\frac{9!}{2!2!}=90720\)। हिंदी प्रश्न में अक्षर गिनते समय शब्द की दी गई स्थिति स्पष्ट देखें। / The English word has (9) letters with (A) twice and (Y) twice, so \(\frac{9!}{2!2!}=90720\). In exams, count repeated letters from the exact word given.
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How many (3)-digit numbers can be formed from (1,3,5,7,9) without repetition?
A. (60)
तीन स्थानों पर क्रम से (5,4,3) विकल्प हैं। कुल \(5\cdot4\cdot3=60\) है। / There are (5,4,3) choices for the three positions. The total is \(5\cdot4\cdot3=60\).
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In how many ways can gold, silver and bronze medals be awarded among (12) runners?
A. (1320)
तीन अलग पदक क्रमित हैं, इसलिए \(^{12}P_3=1320\)। परीक्षा में पदक अलग हों तो permutation प्रयोग करें। / The three medals are ordered, so \(^{12}P_3=1320\). In exams, use permutation when medals are different.
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In how many ways can (6) distinct pictures be hung in a row on a wall?
A. (720)
एक पंक्ति में (6) अलग चित्रों की व्यवस्था (6!=720) है। परीक्षा में row शब्द आए तो linear arrangement लें। / The arrangement of (6) distinct pictures in a row is (6!=720). In exams, the word row means linear arrangement.
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In how many ways can (7) distinct letters be placed one each in (7) distinct envelopes?
A. (5040)
एक-एक मिलान की संख्या (7!=5040) होती है। परीक्षा में distinct envelopes हों तो letters की arrangement करें। / The number of one-to-one placements is (7!=5040). In exams, distinct envelopes mean arranging the letters.
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How many distinct arrangements are possible from the letters of GANANA?
A. (60)
(6) अक्षरों में (3) समान और (2) समान हैं, इसलिए \(\frac{6!}{3!2!}=60\)। परीक्षा में सभी repeated groups का गुणन हर में रखें। / There are (6) letters with (3) identical and (2) identical letters, so \(\frac{6!}{3!2!}=60\). In exams, multiply all repeated group factorials in the denominator.
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A. (30240)
कुल (8!) से साथ बैठने वाली \(7!\cdot2!\) व्यवस्थाएं घटाएं। उत्तर (40320-10080=30240) है। / Subtract the together arrangements \(7!\cdot2!\) from total (8!). The answer is (40320-10080=30240).
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In how many ways can (5) men and (5) women sit alternately in a row?
A. (28800)
दो pattern संभव हैं और प्रत्येक में \(5!\cdot5!\) व्यवस्थाएं हैं। कुल \(2\cdot5!\cdot5!=28800\) है। / There are two possible patterns and each has \(5!\cdot5!\) arrangements. The total is \(2\cdot5!\cdot5!=28800\).
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How many (4)-digit numbers can be formed from (0,1,2,5,7,8) without repetition?
A. (300)
हजार के स्थान पर (0) नहीं आएगा, इसलिए \(5\cdot5\cdot4\cdot3=300\)। परीक्षा में पहले non-zero leading digit चुनें। / The thousands place cannot be (0), so \(5\cdot5\cdot4\cdot3=300\). In exams, first choose the non-zero leading digit.
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A. (2880)
पहले अंग्रेजी पुस्तकें (4!) तरीकों से रखें, फिर (5) gaps में (4) हिंदी पुस्तकें \(^{5}P_4\) तरीकों से रखें। कुल \(4!\cdot^{5}P_4=2880\) है। / Arrange English books first in (4!) ways, then place (4) Hindi books in (5) gaps in \(^{5}P_4\) ways. The total is \(4!\cdot^{5}P_4=2880\).
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In how many ways can a necklace be made with (9) distinct beads if flipping is considered the same?
A. (20160)
माला में घुमाव और पलटना दोनों समान होते हैं, इसलिए (\frac{(9-1)!}{2}=20160)। परीक्षा में necklace और circular table में फर्क रखें। / For a necklace, rotations and reflections are both the same, so (\frac{(9-1)!}{2}=20160). In exams, distinguish necklaces from circular table arrangements.
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A. (120)
कोड में क्रम महत्वपूर्ण होता है, इसलिए \(^{6}P_3=120\)। परीक्षा में code word दिखे तो permutation लगाएं। / Order is important in a code, so \(^{6}P_3=120\). In exams, the word code indicates permutation.
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How many arrangements are possible using all letters of SAFAL?
A. (60)
(5) अक्षरों में (2) समान हैं, इसलिए \(\frac{5!}{2!}=60\)। परीक्षा में repeated vowel को ध्यान से देखें। / There are (5) letters with (2) identical letters, so \(\frac{5!}{2!}=60\). In exams, check repeated vowels carefully.
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What is the number of ways to arrange (4) objects selected from (10) distinct objects in a row?
A. (5040)
यह \(^{10}P_4=10\cdot9\cdot8\cdot7=5040\) है। परीक्षा में selected and arranged हो तो \(nP_r\) लगाएं। / This is \(^{10}P_4=10\cdot9\cdot8\cdot7=5040\). In exams, selected and arranged means use \(nP_r\).
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A. (360)
गोल माला के लिए संख्या (\frac{(7-1)!}{2}=360) है। परीक्षा में garland में reflection को भी समान मानें। / For a circular garland, the number is (\frac{(7-1)!}{2}=360). In exams, count reflection as the same for garlands.
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How many (3)-digit numbers greater than (400) can be formed from (1,2,4,6,8) without repetition?
A. (36)
सैकड़े के स्थान पर (4,6,8) आ सकते हैं और बाकी \(4\cdot3\) तरीके हैं। कुल \(3\cdot4\cdot3=36\) है। / The hundreds place can be (4,6,8), and the remaining places have \(4\cdot3\) ways. The total is \(3\cdot4\cdot3=36\).
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A. (10080)
विशेष व्यक्ति के लिए (2) स्थान हैं और बाकी (7!) तरीकों से बैठेंगे। कुल \(2\cdot7!=10080\) है। / The particular person has (2) possible positions and the remaining people sit in (7!) ways. The total is \(2\cdot7!=10080\).
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In how many ways can (6) distinct prizes be distributed one each among (6) students?
A. (720)
एक-एक अलग पुरस्कार बांटना (6!) तरीकों से होगा। परीक्षा में distinct prizes और one each हो तो factorial लगाएं। / Distributing one distinct prize to each student can be done in (6!) ways. In exams, distinct prizes with one each mean factorial.
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How many distinct arrangements can be made from the letters of ANUPAT?
A. (360)
(6) अक्षरों में (2) समान हैं, इसलिए \(\frac{6!}{2!}=360\)। परीक्षा में repeated letter को denominator में रखें। / There are (6) letters with (2) identical letters, so \(\frac{6!}{2!}=360\). In exams, put the repeated letter factorial in the denominator.
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A. (14400)
पहले लड़कों को (5!) तरीकों से बैठाएं, फिर (6) खाली स्थानों में (3) लड़कियां \(^{6}P_3\) तरीकों से बैठेंगी। कुल \(5!\cdot^{6}P_3=14400\) है। / Arrange boys first in (5!) ways, then place (3) girls in (6) gaps in \(^{6}P_3\) ways. The total is \(5!\cdot^{6}P_3=14400\).
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How many (4)-digit numbers can be formed from (1,2,3,4,5,6,7) without repetition and ending in (5)?
A. (120)
अंतिम अंक निश्चित (5) है और बाकी तीन स्थान \(6\cdot5\cdot4\) तरीकों से भरेंगे। कुल (120) है। / The last digit is fixed as (5), and the other three places are filled in \(6\cdot5\cdot4\) ways. The total is (120).
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A. (48)
दो विशेष किताबें सिरों पर (2!) तरीकों से और बाकी (4!) तरीकों से रखी जाएंगी। कुल \(2!\cdot4!=48\) है। / The two particular books occupy the ends in (2!) ways and the remaining books in (4!) ways. The total is \(2!\cdot4!=48\).
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A. (3600)
विशेष व्यक्ति के लिए (5) अंदरूनी स्थान हैं और बाकी (6!) तरीकों से बैठेंगे। कुल \(5\cdot6!=3600\) है। / The particular person has (5) inner positions and the others sit in (6!) ways. The total is \(5\cdot6!=3600\).
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How many distinct arrangements can be made from the letters of SAMBHAVNA?
A. (181440)
(9) अक्षरों में (2) अक्षर दो-दो बार आते हैं, इसलिए \(\frac{9!}{2!2!}=90720\); सही गणना में repeated letters को ठीक गिनना जरूरी है। / There are (9) letters with two different letters repeated twice, so \(\frac{9!}{2!2!}=90720\). In exams, accurate repeated-letter counting is essential.
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In how many ways can (4) students be called in speaking order from (8) distinct students?
A. (1680)
भाषण का क्रम महत्वपूर्ण है, इसलिए \(^{8}P_4=1680\)। परीक्षा में order of performance को permutation मानें। / The speaking order is important, so \(^{8}P_4=1680\). In exams, order of performance means permutation.
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How many (4)-digit odd numbers can be formed from (0,1,3,5,7,9) without repetition?
A. (500)
इकाई स्थान पर (5) विषम विकल्प हैं, लेकिन शेष में शुरुआत (0) न हो इसका ध्यान चाहिए। कुल \(5\cdot4\cdot4\cdot3=240\) है। / There are (5) odd choices for the unit place, but the first digit must not be (0). The total is \(5\cdot4\cdot4\cdot3=240\).
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