
question_answer1) If \[n\] is even and the value of \[^{n}{{C}_{r}}\] is maximum, then \[r=\]
A) \[\frac{n}{2}\]
B) \[\frac{n+1}{2}\]
C) \[\frac{n1}{2}\]
D) None of these
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question_answer2) A man has 7 friends. In how many ways he can invite one or more of them for a tea party
A) 128
B) 256
C) 127
D) 130
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question_answer3) There are 12 volleyball players in all in a college, out of which a team of 9 players is to be formed. If the captain always remains the same, then in how many ways can the team be formed
A) 36
B) 108
C) 99
D) 165
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question_answer4) In how many ways can a girl and a boy be selected from a group of 15 boys and 8 girls
A) \[15\times 8\]
B) \[15+8\]
C) \[^{23}{{P}_{2}}\]
D) \[^{23}{{C}_{2}}\]
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question_answer5) If \[^{15}{{C}_{3r}}{{=}^{15}}{{C}_{r+3}}\], then the value of \[r\] is [IIT 1967; RPET 1991; MP PET 1998; Karnataka CET 1996]
A) 3
B) 4
C) 5
D) 8
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question_answer6) \[^{47}{{C}_{4}}+\underset{r=1}{\overset{5}{\mathop{\sum }}}\,{}^{52r}{{C}_{3}}=\] [IIT 1980; RPET 2002; UPSEAT 2000]
A) \[^{47}{{C}_{6}}\]
B) \[^{52}{{C}_{5}}\]
C) \[^{15}{{C}_{15}}\]
D) None of these
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question_answer7) \[^{n}{{C}_{r}}{{\div }^{n}}{{C}_{r1}}=\] [MP PET 1984]
A) \[\frac{nr}{r}\]
B) \[\frac{n+r1}{r}\]
C) \[\frac{nr+1}{r}\]
D) \[\frac{nr1}{r}\]
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question_answer8) If \[^{2n}{{C}_{3}}:{{\,}^{n}}{{C}_{2}}=44:3\], then for which of the following values of \[r\], the value of \[^{n}{{C}_{r}}\] will be 15 [MP PET 1981]
A) \[r=3\]
B) \[r=4\]
C) \[r=6\]
D) \[r=5\]
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question_answer9) If \[2\times {}^{n}{{C}_{5}}=9\,\,\times \,\,{}^{n2}{{C}_{5}}\], then the value of n will be
A) 7
B) 10
C) 9
D) 5
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question_answer10) If \[^{{{n}^{2}}n}{{C}_{2}}{{=}^{{{n}^{2}}n}}{{C}_{10}}\], then \[n=\]
A) 12
B) 4 only
C) \[3\]only
D) 4 or \[3\]
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question_answer11) If \[^{n}{{C}_{r1}}=36,{{\ }^{n}}{{C}_{r}}=84\] and \[^{n}{{C}_{r+1}}=126\], then the value of \[r\] is [IIT 1979; Pb. CET 1993, 2003; DCE 1999, 2000; MP PET 2001]
A) 1
B) 2
C) 3
D) None of these
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question_answer12) \[^{n}{{C}_{r}}+{{2}^{n}}{{C}_{r1}}{{+}^{n}}{{C}_{r2}}=\]
A) \[^{n+1}{{C}_{r}}\]
B) \[^{n+1}{{C}_{r+1}}\]
C) \[^{n+2}{{C}_{r}}\]
D) \[^{n+2}{{C}_{r+1}}\]
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question_answer13) In a conference of 8 persons, if each person shake hand with the other one only, then the total number of shake hands shall be [MP PET 1984]
A) 64
B) 56
C) 49
D) 28
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question_answer14) \[^{n}{{C}_{r}}{{+}^{n}}{{C}_{r1}}\] is equal to [MP PET 1984; Kerala (Engg.) 2002]
A) \[^{n+1}{{C}_{r}}\]
B) \[^{n}{{C}_{r+1}}\]
C) \[^{n+1}{{C}_{r+1}}\]
D) \[^{n1}{{C}_{r1}}\]
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question_answer15) If \[^{8}{{C}_{r}}{{=}^{8}}{{C}_{r+2}}\], then the value of \[^{r}{{C}_{2}}\] is [MP PET 1984; RPET 1987]
A) 8
B) 3
C) 5
D) 2
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question_answer16) If \[^{20}{{C}_{n+2}}{{=}^{n}}{{C}_{16}}\], then the value of \[n\] is [MP PET 1984]
A) 7
B) 10
C) 13
D) No value
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question_answer17) The value of \[^{15}{{C}_{3}}{{+}^{15}}{{C}_{13}}\] is [MP PET 1983]
A) \[^{16}{{C}_{3}}\]
B) \[^{30}{{C}_{16}}\]
C) \[^{15}{{C}_{10}}\]
D) \[^{15}{{C}_{15}}\]
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question_answer18) Everybody in a room shakes hand with everybody else. The total number of hand shakes is 66. The total number of persons in the room is [MNR 1991; Kurukshetra CEE 1998; Kerala (Engg.) 2001]
A) 11
B) 12
C) 13
D) 14
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question_answer19) The solution set of \[^{10}{{C}_{x1}}>2\ .{{\ }^{10}}{{C}_{x}}\] is
A) {1, 2, 3}
B) {4, 5, 6}
C) {8,9, 10}
D) {9, 10, 11}
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question_answer20) \[\sum\limits_{r=0}^{m}{^{n+r}{{C}_{n}}=}\] [Pb. CET 2003]
A) \[^{n+m+1}{{C}_{n+1}}\]
B) \[^{n+m+2}{{C}_{n}}\]
C) \[^{n+m+3}{{C}_{n1}}\]
D) None of these
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question_answer21) In a football championship, there were played 153 matches. Every team played one match with each other. The number of teams participating in the championship is [WB JEE 1992; Kurukshetra CEE 1998]
A) 17
B) 18
C) 9
D) 13
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question_answer22) In an examination there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answers correct, is [Pb. CET 1990; UPSEAT 2001]
A) 11
B) 12
C) 27
D) 63
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question_answer23) If \[\alpha {{=}^{m}}{{C}_{2}}\], then \[^{\alpha }{{C}_{2}}\]is equal to
A) \[^{m+1}{{C}_{4}}\]
B) \[^{m1}{{C}_{4}}\]
C) \[3\,.{{\ }^{m+2}}{{C}_{4}}\]
D) \[3\ .{{\ }^{m+1}}{{C}_{4}}\]
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question_answer24) On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are 20 students in the class, then the total number of greeting cards exchanged by the students is
A) \[^{20}{{C}_{2}}\]
B) \[2\ .{{\ }^{20}}{{C}_{2}}\]
C) \[2\ .{{\ }^{20}}{{P}_{2}}\]
D) None of these
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question_answer25) In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is
A) \[{{2}^{32}}\]
B) \[{{(32)}^{2}}1\]
C) \[{{2}^{32}}1\]
D) \[{{2}^{321}}\]
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question_answer26) If \[^{2n}{{C}_{2}}{{:}^{n}}{{C}_{2}}=9:2\] and \[^{n}{{C}_{r}}=10\], then \[r=\]
A) 1
B) 2
C) 4
D) 5
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question_answer27) If \[^{10}{{C}_{r}}{{=}^{10}}{{C}_{r+2}}\], then \[^{5}{{C}_{r}}\] equals [RPET 1996]
A) 120
B) 10
C) 360
D) 5
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question_answer28) If \[^{n}{{C}_{r}}=84,{{\ }^{n}}{{C}_{r1}}=36\] and \[^{n}{{C}_{r+1}}=126\], then \[n\] equals [RPET 1997; MP PET 2001]
A) 8
B) 9
C) 10
D) 5
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question_answer29) If \[^{n}{{C}_{3}}+{{\,}^{n}}{{C}_{4}}>{{\,}^{n+1}}{{C}_{3}},\]then [RPET 1999]
A) \[n>6\]
B) \[n>7\]
C) \[n<6\]
D) None of these
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question_answer30) Value of r for which \[^{15}{{C}_{r+3}}={{\,}^{15}}{{C}_{2r6}}\] is [Pb. CET 1999]
A) 2
B) 4
C) 6
D)  9
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question_answer31) If \[^{n+1}{{C}_{3}}=2{{\,}^{n}}{{C}_{2}},\] then n = [MP PET 2000; Pb. CET 2002]
A) 3
B) 4
C) 5
D) 6
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question_answer32) \[\left( \begin{matrix} n \\ nr \\ \end{matrix} \right)\,+\,\left( \begin{matrix} n \\ r+1 \\ \end{matrix} \right)\], whenever \[0\le r\le n1\]is equal to [AMU 2000]
A) \[\left( \begin{matrix} n \\ r1 \\ \end{matrix} \right)\]
B) \[\left( \begin{matrix} n \\ r \\ \end{matrix} \right)\]
C) \[\left( \begin{matrix} n \\ r+1 \\ \end{matrix} \right)\]
D) \[\left( \begin{matrix} n+1 \\ r+1 \\ \end{matrix} \right)\]
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question_answer33) The least value of natural number n satisfying \[C(n,\,5)+C(n,\,6)\,\,>C(n+1,\,5)\] is [EAMCET 2002]
A) 11
B) 10
C) 12
D) 13
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question_answer34) There are 15 persons in a party and each person shake hand with another, then total number of hand shakes is [RPET 2002]
A) \[^{15}{{P}_{2}}\]
B) \[^{15}{{C}_{2}}\]
C) \[15\,!\]
D) \[2\,(15\,!)\]
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question_answer35) If \[n\] and \[r\] are two positive integers such that \[n\ge r,\] then \[^{n}{{C}_{r1}}\]\[+{{\,}^{n}}{{C}_{r}}=\] [Kerala (Engg.) 2002]
A) \[^{n}{{C}_{nr}}\]
B) \[^{n}{{C}_{r}}\]
C) \[^{n1}{{C}_{r}}\]
D) \[^{n+1}{{C}_{r}}\]
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question_answer36) If \[^{43}{{C}_{r6}}={{\,}^{43}}{{C}_{3r+1}},\] then the value of r is [Kerala (Engg.) 2002]
A) 12
B) 8
C) 6
D) 10
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question_answer37) How many numbers of 6 digits can be formed from the digits of the number 112233 [Karnataka CET 2004]
A) 30
B) 60
C) 90
D) 120
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question_answer38) In an election there are 8 candidates, out of which 5 are to be choosen. If a voter may vote for any number of candidates but not greater than the number to be choosen, then in how many ways can a voter vote
A) 216
B) 114
C) 218
D) None of these
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question_answer39) In an election the number of candidates is 1 greater than the persons to be elected. If a voter can vote in 254 ways, then the number of candidates is
A) 7
B) 10
C) 8
D) 6
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question_answer40) In how many ways can 21 English and 19 Hindi books be placed in a row so that no two Hindi books are together
A) 1540
B) 1450
C) 1504
D) 1405
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question_answer41) \[^{n}{{C}_{r}}{{+}^{n1}}{{C}_{r}}+......{{+}^{r}}{{C}_{r}}\] = [AMU 2002]
A) \[^{n+1}{{C}_{r}}\]
B) \[^{n+1}{{C}_{r+1}}\]
C) \[^{n+2}{{C}_{r}}\]
D) \[{{2}^{n}}\]
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question_answer42) How many words can be formed by taking 3 consonants and 2 vowels out of 5 consonants and 4 vowels
A) \[^{5}{{C}_{3}}\times {{\,}^{4}}{{C}_{2}}\]
B) \[\frac{^{5}{{C}_{3}}\times {{\,}^{4}}{{C}_{2}}}{5}\]
C) \[^{5}{{C}_{3}}\times {{\,}^{4}}{{C}_{3}}\]
D) \[{{(}^{5}}{{C}_{3}}\times {{\,}^{4}}{{C}_{2}})\,(5)\,!\]
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question_answer43) In how many ways a team of 11 players can be formed out of 25 players, if 6 out of them are always to be included and 5 are always to be excluded
A) 2020
B) 2002
C) 2008
D) 8002
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question_answer44) In how many ways can a committee consisting of one or more members be formed out of 12 members of the Municipal Corporation
A) 4095
B) 5095
C) 4905
D) 4090
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question_answer45) Out of 10 white, 9 black and 7 red balls, the number of ways in which selection of one or more balls can be made, is
A) 881
B) 891
C) 879
D) 892
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question_answer46) The numbers of permutations of \[n\] things taken \[r\] at a time, when \[p\]things are always included, is
A) \[^{n}{{C}_{r}}\ p\ !\]
B) \[^{np}{{C}_{r}}\ r\ !\]
C) \[^{np}{{C}_{rp}}\ r\ !\]
D) None of these
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question_answer47) Two packs of 52 cards are shuffled together. The number of ways in which a man can be dealt 26 cards so that he does not get two cards of the same suit and same denomination is
A) \[^{52}{{C}_{26}}\ .\ {{2}^{26}}\]
B) \[^{104}{{C}_{26}}\]
C) \[2\ .{{\ }^{52}}{{C}_{26}}\]
D) None of these
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question_answer48) In a touring cricket team there are 16 players in all including 5 bowlers and 2 wicketkeepers. How many teams of 11 players from these, can be chosen, so as to include three bowlers and one wicketkeeper [MP PET 1984]
A) 650
B) 720
C) 750
D) 800
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question_answer49) Out of 6 books, in how many ways can a set of one or more books be chosen [MP PET 1984]
A) 64
B) 63
C) 62
D) 65
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question_answer50) Choose the correct number of ways in which 15 different books can be divided into five heaps of equal number of books [MP PET 1982]
A) \[\frac{15\ !}{5\ !\ {{(3\ !)}^{5}}}\]
B) \[\frac{15\ !}{{{(3\ !)}^{5}}}\]
C) \[^{15}{{C}_{5}}\]
D) \[^{15}{{P}_{5}}\]
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question_answer51) The number of ways of dividing 52 cards amongst four players equally, are [IIT 1979]
A) \[\frac{52\ !}{{{(13\ !)}^{4}}}\]
B) \[\frac{52\ !}{{{(13\ !)}^{2}}\ 4\ !}\]
C) \[\frac{52\ !}{{{(12\ !)}^{4}}\ (4\ !)}\]
D) None of these
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question_answer52) How many words of 4 consonants and 3 vowels can be formed from 6 consonants and 5 vowels [RPET 1985]
A) 75000
B) 756000
C) 75600
D) None of these
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question_answer53) In the 13 cricket players 4 are bowlers, then how many ways can form a cricket team of 11 players in which at least 2 bowlers included [RPET 1988]
A) 55
B) 72
C) 78
D) None of these
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question_answer54) Six '+' and four '' signs are to placed in a straight line so that no two '' signs come together, then the total number of ways are [IIT 1988]
A) 15
B) 18
C) 35
D) 42
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question_answer55) The number of groups that can be made from 5 different green balls, 4 different blue balls and 3 different red balls, if at least 1 green and 1 blue ball is to be included [IIT 1974]
A) 3700
B) 3720
C) 4340
D) None of these
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question_answer56) In how many ways can 6 persons be selected from 4 officers and 8 constables, if at least one officer is to be included [Roorkee 1985; MP PET 2001]
A) 224
B) 672
C) 896
D) None of these
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question_answer57) To fill 12 vacancies there are 25 candidates of which five are from scheduled caste. If 3 of the vacancies are reserved for scheduled caste candidates while the rest are open to all, then the number of ways in which the selection can be made [RPET 1981]
A) \[^{5}{{C}_{3}}{{\times }^{22}}{{C}_{9}}\]
B) \[^{22}{{C}_{9}}{{}^{5}}{{C}_{3}}\]
C) \[^{22}{{C}_{3}}{{+}^{5}}{{C}_{3}}\]
D) None of these
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question_answer58) In an election there are 5 candidates and three vacancies. A voter can vote maximum to three candidates, then in how many ways can he vote [MP PET 1987]
A) 125
B) 60
C) 10
D) 25
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question_answer59) There are 9 chairs in a room on which 6 persons are to be seated, out of which one is guest with one specific chair. In how many ways they can sit [MP PET 1987]
A) 6720
B) 60480
C) 30
D) 346
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question_answer60) Out of 6 boys and 4 girls, a group of 7 is to be formed. In how many ways can this be done if the group is to have a majority of boys [MP PET 1994]
A) 120
B) 90
C) 100
D) 80
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question_answer61) The number of ways in which 10 persons can go in two boats so that there may be 5 on each boat, supposing that two particular persons will not go in the same boat is [Pb. CET 1999]
A) \[\frac{1}{2}{{(}^{10}}{{C}_{5}})\]
B) \[2{{(}^{8}}{{C}_{4}})\]
C) \[\frac{1}{2}{{(}^{8}}{{C}_{5}})\]
D) None of these
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question_answer62) The number of ways in which we can select three numbers from 1 to 30 so as to exclude every selection of all even numbers is
A) 4060
B) 3605
C) 455
D) None of these
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question_answer63) A total number of words which can be formed out of the letters \[a,\ b,\ c,\ d,\ e,\ f\] taken 3 together such that each word contains at least one vowel, is
A) 72
B) 48
C) 96
D) None of these
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question_answer64) The number of ways in which any four letters can be selected from the word ?CORGOO? is
A) 15
B) 11
C) 7
D) None of these
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question_answer65) The total number of natural numbers of six digits that can be made with digits 1, 2, 3, 4, if all digits are to appear in the same number at least once, is
A) 1560
B) 840
C) 1080
D) 480
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question_answer66) All possible two factors products are formed from numbers 1, 2, 3, 4, ...., 200. The number of factors out of the total obtained which are multiples of 5 is
A) 5040
B) 7180
C) 8150
D) None of these
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question_answer67) The total number of ways of selecting six coins out of 20 one rupee coins, 10 fifty paise coins and 7 twenty five paise coins is
A) 28
B) 56
C) \[^{37}{{C}_{6}}\]
D) None of these
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question_answer68) The number of ways in which thirty five apples can be distributed among 3 boys so that each can have any number of apples, is
A) 1332
B) 666
C) 333
D) None of these
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question_answer69) A father with 8 children takes them 3 at a time to the Zoological gardens, as often as he can without taking the same 3 children together more than once. The number of times he will go to the garden is
A) 336
B) 112
C) 56
D) None of these
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question_answer70) In how many ways can 5 red and 4 white balls be drawn from a bag containing 10 red and 8 white balls [EAMCET 1991; Pb. CET 2000]
A) \[^{8}{{C}_{5}}{{\times }^{10}}{{C}_{4}}\]
B) \[^{10}{{C}_{5}}{{\times }^{8}}{{C}_{4}}\]
C) \[^{18}{{C}_{9}}\]
D) None of these
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question_answer71) \[^{14}{{C}_{4}}+\sum\limits_{j=1}^{4}{^{18j}{{C}_{3}}}\] is equal to [EAMCET 1991]
A) \[^{18}{{C}_{3}}\]
B) \[^{18}{{C}_{4}}\]
C) \[^{14}{{C}_{7}}\]
D) None of these
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question_answer72) The number of ways in which four letters of the word 'MATHEMATICS' can be arranged is given by [Kurukshetra CEE 1996; Pb. CET 1995]
A) 136
B) 192
C) 1680
D) 2454
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question_answer73) 10 different letters of English alphabet are given. Out of these letters, words of 5 letters are formed. How many words are formed when at least one letter is repeated [UPSEAT 1999]
A) 99748
B) 98748
C) 96747
D) 97147
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question_answer74) The number of ways in which a committee of 6 members can be formed from 8 gentlemen and 4 ladies so that the committee contains at least 3 ladies is [Kerala (Engg.) 2002]
A) 252
B) 672
C) 444
D) 420
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question_answer75) A person is permitted to select at least one and at most n coins from a collection of \[(2n+1)\] distinct coins. If the total number of ways in which he can select coins is 255, then n equals [AMU 2002]
A) 4
B) 8
C) 16
D) 32
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question_answer76) A man has 10 friends. In how many ways he can invite one or more of them to a party [AMU 2002]
A) \[10\,!\]
B) \[{{2}^{10}}\]
C) \[10\,!\,\,1\]
D) \[{{2}^{10}}1\]
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question_answer77) A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five question. The number of choices available to him is [AIEEE 2003]
A) 140
B) 196
C) 280
D) 346
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question_answer78) If \[^{n}{{C}_{r}}\] denotes the number of combinations of n things taken r at a time, then the expression \[^{n}{{C}_{r+1}}+{{\,}^{n}}{{C}_{r1}}+\,2\times {{\,}^{n}}{{C}_{r}}\] equals [AIEEE 2003]
A) \[^{n+2}{{C}_{r}}\]
B) \[^{n+2}{{C}_{r+1}}\]
C) \[^{n+1}{{C}_{r}}\]
D) \[^{n+1}{{C}_{r+1}}\]
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question_answer79) A student is allowed to select at most \[n\] books from a collection of \[(2n+1)\] books. If the total number of ways in which he can select one book is 63, then the value of \[n\] is [IIT 1987; RPET 1999; Pb. CET 2003; Orissa JEE 2005]
A) 2
B) 3
C) 4
D) None of these
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question_answer80) \[^{n1}{{C}_{r}}=({{k}^{2}}3)\,.{{\,}^{n}}{{C}_{r+1}}\] if \[k\in \] [IIT Screening 2004]
A) \[[\sqrt{3},\,\sqrt{3}]\]
B) \[(\infty ,\,2)\]
C) \[(2,\,\infty )\]
D) \[(\sqrt{3},\,2)\]
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question_answer81) The value of \[\sum\limits_{r=0}^{n1}{\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r}}+{{\,}^{n}}{{C}_{r+1}}}}\] equals [MP PET 2004]
A) \[n+1\]
B) \[\frac{n}{2}\]
C) \[n+2\]
D) None of these
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question_answer82) Out of 5 apples, 10 mangoes and 15 oranges, any 15 fruits distributed among two persons. The total number of ways of distribution [DCE 2005]
A) 66
B) 36
C) 60
D) None of these
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question_answer83) The value of \[{}^{50}{{C}_{4}}+\sum\limits_{r=1}^{6}{^{56r}{{C}_{3}}}\] is [AIEEE 2005]
A) \[^{56}{{C}_{3}}\]
B) \[^{56}{{C}_{4}}\]
C) \[^{55}{{C}_{4}}\]
D) \[^{55}{{C}_{3}}\]
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question_answer84) If \[^{n}{{C}_{12}}={{\,}^{n}}{{C}_{6}}\], then \[^{n}{{C}_{2}}=\] [Karnataka CET 2005]
A) 72
B) 153
C) 306
D) 2556
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question_answer85) A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is [Kerala (Engg.) 2005]
A) 140
B) 196
C) 280
D) 346
E) 265
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