If both are included there are \(\binom{9}{3}=84\) ways and if both are excluded there are \(\binom{9}{5}=126\) ways. The total is (210).
Step 2
Why this answer is correct
The correct answer is C. (280). If both are included there are \(\binom{9}{3}=84\) ways and if both are excluded there are \(\binom{9}{5}=126\) ways. The total is (210).
Step 3
Exam Tip
दोनों शामिल हों तो \(\binom{9}{3}=84\) और दोनों बाहर हों तो \(\binom{9}{5}=126\) तरीके हैं। कुल (84+126=210) है।
For a failed triangle (3) points are chosen from the (7) collinear points. So the number is \(\binom{7}{3}\).
Step 2
Why this answer is correct
The correct answer is B. \(\binom{7}{3}\). For a failed triangle (3) points are chosen from the (7) collinear points. So the number is \(\binom{7}{3}\).
Step 3
Exam Tip
असफल त्रिभुज के लिए (7) समरेखीय बिंदुओं में से (3) चुने जाते हैं। इसलिए संख्या \(\binom{7}{3}\) है।
The (2) numbers are fixed and (2) are excluded. The remaining (3) numbers are chosen from (9) in \(\binom{9}{3}=84\) ways.
Step 2
Why this answer is correct
The correct answer is A. (84). The (2) numbers are fixed and (2) are excluded. The remaining (3) numbers are chosen from (9) in \(\binom{9}{3}=84\) ways.
Step 3
Exam Tip
(2) संख्याएं तय हैं और (2) हट गई हैं। बाकी (3) संख्याएं (9) में से \(\binom{9}{3}=84\) तरीकों से चुनी जाएंगी।
Total ways are \(\binom{8}{4}=70\) and if both special subjects are included there are \(\binom{6}{2}=15\) ways. Hence (70-15=55).
Step 2
Why this answer is correct
The correct answer is C. (55). Total ways are \(\binom{8}{4}=70\) and if both special subjects are included there are \(\binom{6}{2}=15\) ways. Hence (70-15=55).
Step 3
Exam Tip
कुल \(\binom{8}{4}=70\) हैं और दोनों विशेष विषय साथ हों तो \(\binom{6}{2}=15\) हैं। इसलिए (70-15=55) है।
The letter (a) is fixed and (b) is excluded so choose the remaining (3) letters from (7). The ways are \(\binom{7}{3}=35\).
Step 2
Why this answer is correct
The correct answer is C. (35). The letter (a) is fixed and (b) is excluded so choose the remaining (3) letters from (7). The ways are \(\binom{7}{3}=35\).
Step 3
Exam Tip
(a) तय है और (b) हट गया है इसलिए बाकी (3) अक्षर (7) में से चुनेंगे। तरीके \(\binom{7}{3}=35\) हैं।
Total ways are \(\binom{10}{5}=252\) and if both special friends do not go then \(\binom{8}{5}=56\). Hence (252-56=196).
Step 2
Why this answer is correct
The correct answer is B. (182). Total ways are \(\binom{10}{5}=252\) and if both special friends do not go then \(\binom{8}{5}=56\). Hence (252-56=196).
Step 3
Exam Tip
कुल \(\binom{10}{5}=252\) हैं और दोनों विशेष न जाएं तो \(\binom{8}{5}=56\) हैं। इसलिए (252-56=196) है।
Total ways are \(\binom{15}{5}=3003\). Removing all-white \(\binom{7}{5}=21\) and all-black \(\binom{8}{5}=56\) gives (2926).
Step 2
Why this answer is correct
The correct answer is A. (2926). Total ways are \(\binom{15}{5}=3003\). Removing all-white \(\binom{7}{5}=21\) and all-black \(\binom{8}{5}=56\) gives (2926).
Step 3
Exam Tip
कुल \(\binom{15}{5}=3003\) हैं। सभी सफेद \(\binom{7}{5}=21\) और सभी काली \(\binom{8}{5}=56\) हटाने पर (2926) मिलते हैं।
Total ways are \(\binom{18}{6}=18564\). Removing the cases of (0) teacher and (1) teacher gives \(18564-\binom{10}{6}-\binom{8}{1}\binom{10}{5}=16338\).
Step 2
Why this answer is correct
The correct answer is B. (17472). Total ways are \(\binom{18}{6}=18564\). Removing the cases of (0) teacher and (1) teacher gives \(18564-\binom{10}{6}-\binom{8}{1}\binom{10}{5}=16338\).
Step 3
Exam Tip
कुल \(\binom{18}{6}=18564\) हैं। (0) शिक्षक और (1) शिक्षक के मामले हटाने पर \(18564-\binom{10}{6}-\binom{8}{1}\binom{10}{5}=16338\) है।
Total subsets are \(\binom{10}{5}=252\) and those containing both (2), (3) are \(\binom{8}{3}=56\). Hence (252-56=196).
Step 2
Why this answer is correct
The correct answer is C. (196). Total subsets are \(\binom{10}{5}=252\) and those containing both (2), (3) are \(\binom{8}{3}=56\). Hence (252-56=196).
Step 3
Exam Tip
कुल \(\binom{10}{5}=252\) हैं और (2), (3) दोनों हों तो \(\binom{8}{3}=56\) हैं। इसलिए (252-56=196) है।
The element (1) is already chosen so the remaining (4) elements are chosen from (9). The number of ways is \(\binom{9}{4}=126\).
Step 2
Why this answer is correct
The correct answer is B. (126). The element (1) is already chosen so the remaining (4) elements are chosen from (9). The number of ways is \(\binom{9}{4}=126\).
Step 3
Exam Tip
(1) पहले से चुना है इसलिए बाकी (4) तत्व (9) में से चुने जाएंगे। तरीकों की संख्या \(\binom{9}{4}=126\) है।
By Pascal's identity \(\binom{n}{r}+\binom{n}{r+1}=\binom{n+1}{r+1}\). Hence the answer is \(\binom{11}{5}\).
Step 2
Why this answer is correct
The correct answer is A. \(\binom{11}{5}\). By Pascal's identity \(\binom{n}{r}+\binom{n}{r+1}=\binom{n+1}{r+1}\). Hence the answer is \(\binom{11}{5}\).
Step 3
Exam Tip
पास्कल पहचान से \(\binom{n}{r}+\binom{n}{r+1}=\binom{n+1}{r+1}\) होता है। इसलिए उत्तर \(\binom{11}{5}\) है।
Total ways are \(\binom{15}{5}=3003\). Removing only pens \(\binom{8}{5}=56\) and only pencils \(\binom{7}{5}=21\) gives (2926).
Step 2
Why this answer is correct
The correct answer is C. (2982). Total ways are \(\binom{15}{5}=3003\). Removing only pens \(\binom{8}{5}=56\) and only pencils \(\binom{7}{5}=21\) gives (2926).
Step 3
Exam Tip
कुल \(\binom{15}{5}=3003\) हैं। केवल पेन \(\binom{8}{5}=56\) और केवल पेंसिल \(\binom{7}{5}=21\) हटाने पर (2926) है।
Total ways are \(\binom{14}{6}=3003\) and ways with no special student are \(\binom{10}{6}=210\). Hence (3003-210=2793) ways.
Step 2
Why this answer is correct
The correct answer is A. (2877). Total ways are \(\binom{14}{6}=3003\) and ways with no special student are \(\binom{10}{6}=210\). Hence (3003-210=2793) ways.
Step 3
Exam Tip
कुल \(\binom{14}{6}=3003\) हैं और कोई विशेष न हो तो \(\binom{10}{6}=210\) हैं। इसलिए (3003-210=2793) तरीके हैं।
The cases are (3), (4), and (5) doctors. The total is \(\binom{8}{3}\binom{6}{2}+\binom{8}{4}\binom{6}{1}+\binom{8}{5}=1176\).
Step 2
Why this answer is correct
The correct answer is A. (1176). The cases are (3), (4), and (5) doctors. The total is \(\binom{8}{3}\binom{6}{2}+\binom{8}{4}\binom{6}{1}+\binom{8}{5}=1176\).
Step 3
Exam Tip
मामले (3), (4) और (5) डॉक्टरों के हैं। कुल \(\binom{8}{3}\binom{6}{2}+\binom{8}{4}\binom{6}{1}+\binom{8}{5}=1176\) है।
Total ways are \(\binom{16}{5}=4368\). Removing only English \(\binom{7}{5}=21\) and only Hindi \(\binom{9}{5}=126\) gives (4221) ways.
Step 2
Why this answer is correct
The correct answer is B. (4326). Total ways are \(\binom{16}{5}=4368\). Removing only English \(\binom{7}{5}=21\) and only Hindi \(\binom{9}{5}=126\) gives (4221) ways.
Step 3
Exam Tip
कुल \(\binom{16}{5}=4368\) हैं। केवल अंग्रेजी \(\binom{7}{5}=21\) और केवल हिंदी \(\binom{9}{5}=126\) हटाने पर (4221) तरीके मिलते हैं।
