Total triples are \(\binom{20}{3}=1140\) and failed triples are \(\binom{8}{3}+\binom{5}{3}=66\). Hence (1140-66=1074).
Step 2
Why this answer is correct
The correct answer is B. (1074). Total triples are \(\binom{20}{3}=1140\) and failed triples are \(\binom{8}{3}+\binom{5}{3}=66\). Hence (1140-66=1074).
Step 3
Exam Tip
कुल \(\binom{20}{3}=1140\) त्रिक हैं और असफल त्रिक \(\binom{8}{3}+\binom{5}{3}=66\) हैं। इसलिए (1140-66=1074) है।
The number of men can be (2), (3), or (4). The total is \(\binom{10}{2}\binom{8}{4}+\binom{10}{3}\binom{8}{3}+\binom{10}{4}\binom{8}{2}=15750\).
Step 2
Why this answer is correct
The correct answer is C. (15750). The number of men can be (2), (3), or (4). The total is \(\binom{10}{2}\binom{8}{4}+\binom{10}{3}\binom{8}{3}+\binom{10}{4}\binom{8}{2}=15750\).
Step 3
Exam Tip
पुरुषों की संख्या (2), (3) या (4) हो सकती है। कुल \(\binom{10}{2}\binom{8}{4}+\binom{10}{3}\binom{8}{3}+\binom{10}{4}\binom{8}{2}=15750\) है।
The cases are (2) special and (3) special students. The total is \(\binom{4}{2}\binom{10}{4}+\binom{4}{3}\binom{10}{3}=2100\).
Step 2
Why this answer is correct
The correct answer is C. (2100). The cases are (2) special and (3) special students. The total is \(\binom{4}{2}\binom{10}{4}+\binom{4}{3}\binom{10}{3}=2100\).
Step 3
Exam Tip
मामले (2) विशेष और (3) विशेष के हैं। कुल \(\binom{4}{2}\binom{10}{4}+\binom{4}{3}\binom{10}{3}=2100\) है।
The number of special objects can be (3), (4), (5), or (6). The total is \(\sum_{r=3}^{6}\binom{6}{r}\binom{10}{7-r}=6960\).
Step 2
Why this answer is correct
The correct answer is B. (6960). The number of special objects can be (3), (4), (5), or (6). The total is \(\sum_{r=3}^{6}\binom{6}{r}\binom{10}{7-r}=6960\).
Step 3
Exam Tip
विशेष वस्तुएं (3), (4), (5) या (6) हो सकती हैं। कुल \(\sum_{r=3}^{6}\binom{6}{r}\binom{10}{7-r}=6960\) है।
Choose (1) from (a,b), then choose the remaining (4) so that (c,d) are not both included. The correct expression is (\binom{2}{1}\left\(\binom{10}{4}-\binom{8}{2}\right\)=364).
Step 2
Why this answer is correct
The correct answer is C. (300). Choose (1) from (a,b), then choose the remaining (4) so that (c,d) are not both included. The correct expression is (\binom{2}{1}\left\(\binom{10}{4}-\binom{8}{2}\right\)=364).
Step 3
Exam Tip
(a,b) में से (1) चुनकर बाकी (4) ऐसे चुनें कि (c,d) दोनों साथ न आएं। तरीके (\binom{2}{1}\left\(\binom{10}{4}-\binom{8}{2}\right\)=364) नहीं, सही (\binom{2}{1}\left\(\binom{10}{4}-\binom{8}{2}\right\)=364) है।
The number of special players can be (1) or (2). The total is \(\binom{3}{1}\binom{10}{5}+\binom{3}{2}\binom{10}{4}=1470\).
Step 2
Why this answer is correct
The correct answer is A. (1470). The number of special players can be (1) or (2). The total is \(\binom{3}{1}\binom{10}{5}+\binom{3}{2}\binom{10}{4}=1470\).
Step 3
Exam Tip
विशेष खिलाड़ी (1) या (2) चुने जा सकते हैं। कुल \(\binom{3}{1}\binom{10}{5}+\binom{3}{2}\binom{10}{4}=1470\) है।
(10) विज्ञान और (6) कला विद्यार्थियों में से (6) विद्यार्थी चुनने हैं जिनमें विज्ञान विद्यार्थियों की संख्या कला विद्यार्थियों की संख्या से ठीक (2) अधिक हो। कितने तरीके हैं?
If science students are (x) and arts students are (y), then (x+y=6) and (x=y+2), so (x=4), (y=2). The ways are \(\binom{10}{4}\binom{6}{2}=3150\).
Step 2
Why this answer is correct
The correct answer is B. (3150). If science students are (x) and arts students are (y), then (x+y=6) and (x=y+2), so (x=4), (y=2). The ways are \(\binom{10}{4}\binom{6}{2}=3150\).
Step 3
Exam Tip
यदि विज्ञान (x) और कला (y) हों तो (x+y=6) और (x=y+2), इसलिए (x=4), (y=2)। तरीके \(\binom{10}{4}\binom{6}{2}=3150\) हैं।
Total ways are \(\binom{17}{5}=6188\). Removing only English \(\binom{8}{5}=56\) and only Hindi \(\binom{9}{5}=126\) gives (6006).
Step 2
Why this answer is correct
The correct answer is A. (6006). Total ways are \(\binom{17}{5}=6188\). Removing only English \(\binom{8}{5}=56\) and only Hindi \(\binom{9}{5}=126\) gives (6006).
Step 3
Exam Tip
कुल \(\binom{17}{5}=6188\) हैं। केवल अंग्रेजी \(\binom{8}{5}=56\) और केवल हिंदी \(\binom{9}{5}=126\) हटाने पर (6006) मिलते हैं।
The number of doctors will be (4), (5), or (6). The total is \(\binom{9}{4}\binom{7}{2}+\binom{9}{5}\binom{7}{1}+\binom{9}{6}=3612\).
Step 2
Why this answer is correct
The correct answer is C. (3612). The number of doctors will be (4), (5), or (6). The total is \(\binom{9}{4}\binom{7}{2}+\binom{9}{5}\binom{7}{1}+\binom{9}{6}=3612\).
Step 3
Exam Tip
डॉक्टरों की संख्या (4), (5) या (6) होगी। कुल \(\binom{9}{4}\binom{7}{2}+\binom{9}{5}\binom{7}{1}+\binom{9}{6}=3612\) है।
Total ways are \(\binom{15}{6}=5005\). Removing selections with (0) and (1) special student gives \(5005-\binom{10}{6}-\binom{5}{1}\binom{10}{5}=3535\).
Step 2
Why this answer is correct
The correct answer is A. (3535). Total ways are \(\binom{15}{6}=5005\). Removing selections with (0) and (1) special student gives \(5005-\binom{10}{6}-\binom{5}{1}\binom{10}{5}=3535\).
Step 3
Exam Tip
कुल \(\binom{15}{6}=5005\) हैं। (0) और (1) विशेष वाले चयन हटाने पर \(5005-\binom{10}{6}-\binom{5}{1}\binom{10}{5}=3535\) है।
The number of pens can be (2), (3), or (4). The total is \(\binom{9}{2}\binom{8}{4}+\binom{9}{3}\binom{8}{3}+\binom{9}{4}\binom{8}{2}=10752\).
Step 2
Why this answer is correct
The correct answer is C. (10752). The number of pens can be (2), (3), or (4). The total is \(\binom{9}{2}\binom{8}{4}+\binom{9}{3}\binom{8}{3}+\binom{9}{4}\binom{8}{2}=10752\).
Step 3
Exam Tip
पेन (2), (3) या (4) हो सकते हैं। कुल \(\binom{9}{2}\binom{8}{4}+\binom{9}{3}\binom{8}{3}+\binom{9}{4}\binom{8}{2}=10752\) है।
By Pascal's identity \(\binom{n}{r}+\binom{n}{r+1}=\binom{n+1}{r+1}\). Hence the answer is \(\binom{12}{5}\).
Step 2
Why this answer is correct
The correct answer is B. \(\binom{12}{5}\). By Pascal's identity \(\binom{n}{r}+\binom{n}{r+1}=\binom{n+1}{r+1}\). Hence the answer is \(\binom{12}{5}\).
Step 3
Exam Tip
पास्कल पहचान से \(\binom{n}{r}+\binom{n}{r+1}=\binom{n+1}{r+1}\) होता है। इसलिए उत्तर \(\binom{12}{5}\) है।
The elements (1) and (2) are fixed and (3) is excluded. The remaining (3) elements are chosen from (9), so \(\binom{9}{3}=84\).
Step 2
Why this answer is correct
The correct answer is B. (84). The elements (1) and (2) are fixed and (3) is excluded. The remaining (3) elements are chosen from (9), so \(\binom{9}{3}=84\).
Step 3
Exam Tip
(1) और (2) तय हैं और (3) हट गया है। बाकी (3) तत्व (9) में से चुने जाएंगे इसलिए \(\binom{9}{3}=84\) है।
Total subsets are \(\binom{11}{6}=462\) and those containing both (4), (5) are \(\binom{9}{4}=126\). Hence (462-126=336).
Step 2
Why this answer is correct
The correct answer is C. (336). Total subsets are \(\binom{11}{6}=462\) and those containing both (4), (5) are \(\binom{9}{4}=126\). Hence (462-126=336).
Step 3
Exam Tip
कुल \(\binom{11}{6}=462\) हैं और (4), (5) दोनों हों तो \(\binom{9}{4}=126\) हैं। इसलिए (462-126=336) है।
Total ways are \(\binom{17}{6}=12376\). Removing all-white \(\binom{8}{6}=28\) and all-black \(\binom{9}{6}=84\) gives (12264).
Step 2
Why this answer is correct
The correct answer is B. (12264). Total ways are \(\binom{17}{6}=12376\). Removing all-white \(\binom{8}{6}=28\) and all-black \(\binom{9}{6}=84\) gives (12264).
Step 3
Exam Tip
कुल \(\binom{17}{6}=12376\) हैं। सभी सफेद \(\binom{8}{6}=28\) और सभी काली \(\binom{9}{6}=84\) हटाने पर (12264) मिलते हैं।
The letter (a) is fixed and (b) is excluded. Choose (1) from (c,d) and (3) from the remaining (6), so \(\binom{2}{1}\binom{6}{3}=40\).
Step 2
Why this answer is correct
The correct answer is B. (40). The letter (a) is fixed and (b) is excluded. Choose (1) from (c,d) and (3) from the remaining (6), so \(\binom{2}{1}\binom{6}{3}=40\).
Step 3
Exam Tip
(a) तय है और (b) हट गया है। (c,d) में से (1) और शेष (6) में से (3) चुनेंगे इसलिए \(\binom{2}{1}\binom{6}{3}=40\) है।
Chemistry is fixed so choose the remaining (3) subjects from (8). Removing (6) selections containing both mathematics and physics leaves (50) ways.
Step 2
Why this answer is correct
The correct answer is B. (50). Chemistry is fixed so choose the remaining (3) subjects from (8). Removing (6) selections containing both mathematics and physics leaves (50) ways.
Step 3
Exam Tip
रसायन तय है इसलिए बाकी (3) विषय (8) में से चुनेंगे। गणित और भौतिकी दोनों साथ वाले (6) चयन हटाने पर (50) तरीके बचते हैं।
The (2) numbers are fixed and (2) are excluded. The remaining (4) numbers are chosen from (11) in \(\binom{11}{4}=330\) ways.
Step 2
Why this answer is correct
The correct answer is B. (330). The (2) numbers are fixed and (2) are excluded. The remaining (4) numbers are chosen from (11) in \(\binom{11}{4}=330\) ways.
Step 3
Exam Tip
(2) संख्याएं तय हैं और (2) हट गई हैं। बाकी (4) संख्याएं (11) में से \(\binom{11}{4}=330\) तरीकों से चुनी जाएंगी।
For a failed triangle (3) points are chosen from the (8) collinear points. So the number is \(\binom{8}{3}\).
Step 2
Why this answer is correct
The correct answer is B. \(\binom{8}{3}\). For a failed triangle (3) points are chosen from the (8) collinear points. So the number is \(\binom{8}{3}\).
Step 3
Exam Tip
असफल त्रिभुज के लिए (8) समरेखीय बिंदुओं में से (3) चुने जाते हैं। इसलिए संख्या \(\binom{8}{3}\) है।
If both are included there are \(\binom{10}{4}=210\) ways and if both are excluded there are \(\binom{10}{6}=210\) ways. The total is (420).
Step 2
Why this answer is correct
The correct answer is B. (420). If both are included there are \(\binom{10}{4}=210\) ways and if both are excluded there are \(\binom{10}{6}=210\) ways. The total is (420).
Step 3
Exam Tip
दोनों शामिल हों तो \(\binom{10}{4}=210\) और दोनों बाहर हों तो \(\binom{10}{6}=210\) तरीके हैं। कुल (420) है।
You can choose (3), (4), or (5) from the last (5) questions. The total is \(\binom{5}{3}\binom{7}{4}+\binom{5}{4}\binom{7}{3}+\binom{5}{5}\binom{7}{2}=546\).
Step 2
Why this answer is correct
The correct answer is C. (546). You can choose (3), (4), or (5) from the last (5) questions. The total is \(\binom{5}{3}\binom{7}{4}+\binom{5}{4}\binom{7}{3}+\binom{5}{5}\binom{7}{2}=546\).
Step 3
Exam Tip
अंतिम (5) में से (3), (4) या (5) प्रश्न चुने जा सकते हैं। कुल \(\binom{5}{3}\binom{7}{4}+\binom{5}{4}\binom{7}{3}+\binom{5}{5}\binom{7}{2}=546\) है।
The number of seniors can be (0), (1), or (2). The total is \(\binom{8}{0}\binom{10}{6}+\binom{8}{1}\binom{10}{5}+\binom{8}{2}\binom{10}{4}=8106\).
Step 2
Why this answer is correct
The correct answer is C. (8106). The number of seniors can be (0), (1), or (2). The total is \(\binom{8}{0}\binom{10}{6}+\binom{8}{1}\binom{10}{5}+\binom{8}{2}\binom{10}{4}=8106\).
Step 3
Exam Tip
वरिष्ठों की संख्या (0), (1) या (2) हो सकती है। कुल \(\binom{8}{0}\binom{10}{6}+\binom{8}{1}\binom{10}{5}+\binom{8}{2}\binom{10}{4}=8106\) है।
Choose (1) of the two fixed cards and (4) cards from the remaining (11). The ways are \(\binom{2}{1}\binom{11}{4}=660\).
Step 2
Why this answer is correct
The correct answer is B. (660). Choose (1) of the two fixed cards and (4) cards from the remaining (11). The ways are \(\binom{2}{1}\binom{11}{4}=660\).
Step 3
Exam Tip
दो निश्चित कार्डों में से (1) चुनें और बाकी (4) कार्ड (11) में से चुनें। तरीके \(\binom{2}{1}\binom{11}{4}=660\) हैं।
(11) खिलाड़ियों में से (5) खिलाड़ियों का चयन करना है। एक कप्तान पहले से तय है और उसे चुना नहीं जाना है लेकिन उपकप्तान अवश्य चुना जाना है। कितने तरीके हैं?
The captain is excluded and the vice-captain is fixed. The remaining (4) players are chosen from (9), so \(\binom{9}{4}=126\).
Step 2
Why this answer is correct
The correct answer is B. (126). The captain is excluded and the vice-captain is fixed. The remaining (4) players are chosen from (9), so \(\binom{9}{4}=126\).
Step 3
Exam Tip
कप्तान हट गया और उपकप्तान तय है। बाकी (4) खिलाड़ी (9) में से चुने जाएंगे इसलिए \(\binom{9}{4}=126\) है।
The number of special objects can be (0), (1), or (2). The total is \(\binom{5}{0}\binom{11}{6}+\binom{5}{1}\binom{11}{5}+\binom{5}{2}\binom{11}{4}=6072\).
Step 2
Why this answer is correct
The correct answer is C. (6072). The number of special objects can be (0), (1), or (2). The total is \(\binom{5}{0}\binom{11}{6}+\binom{5}{1}\binom{11}{5}+\binom{5}{2}\binom{11}{4}=6072\).
Step 3
Exam Tip
विशेष वस्तुएं (0), (1) या (2) चुनी जा सकती हैं। कुल \(\binom{5}{0}\binom{11}{6}+\binom{5}{1}\binom{11}{5}+\binom{5}{2}\binom{11}{4}=6072\) है।
