8666 + Mcqs in Mathematics Page 124 McqOptions

6151.	If \[^{n}{{C}_{12}}={{\,}^{n}}{{C}_{6}}\], then \[^{n}{{C}_{2}}=\] [Karnataka CET 2005]
A.	72
B.	153
C.	306
D.	2556
Answer» C. 306

Discussion

6152.	The value of \[{}^{50}{{C}_{4}}+\sum\limits_{r=1}^{6}{^{56-r}{{C}_{3}}}\] is [AIEEE 2005]
A.	\[^{56}{{C}_{3}}\]
B.	\[^{56}{{C}_{4}}\]
C.	\[^{55}{{C}_{4}}\]
D.	\[^{55}{{C}_{3}}\]
Answer» C. \[^{55}{{C}_{4}}\]

Discussion

6153.	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
Answer» B. 36

Discussion

6154.	The value of \[\sum\limits_{r=0}^{n-1}{\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r}}+{{\,}^{n}}{{C}_{r+1}}}}\] equals [MP PET 2004]
A.	\[n+1\]
B.	\[\frac{n}{2}\]
C.	\[n+2\]
D.	None of these
Answer» C. \[n+2\]

Discussion

6155.	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.	\[r=5\]
Answer» C. \[r=6\]

Discussion

6156.	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
Answer» C. 4

Discussion

6157.	If \[^{n}{{C}_{r}}\] denotes the number of combinations of n things taken r at a time, then the expression \[^{n}{{C}_{r+1}}+{{\,}^{n}}{{C}_{r-1}}+\,2\times {{\,}^{n}}{{C}_{r}}\] equals [AIEEE 2003]
A.	\[^{n+2}{{C}_{r}}\]
B.	\[^{n+2}{{C}_{r+1}}\]
C.	\[^{n+1}{{C}_{r}}\]
D.	\[^{n+1}{{C}_{r+1}}\]
Answer» C. \[^{n+1}{{C}_{r}}\]

Discussion

6158.	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.	\[{{2}^{10}}-1\]
Answer» E.

Discussion

6159.	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
Answer» B. 8

Discussion

6160.	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
Answer» B. 98748

Discussion

6161.	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
Answer» E.

Discussion

6162.	\[^{14}{{C}_{4}}+\sum\limits_{j=1}^{4}{^{18-j}{{C}_{3}}}\] is equal to [EAMCET 1991]
A.	\[^{18}{{C}_{3}}\]
B.	\[^{18}{{C}_{4}}\]
C.	\[^{14}{{C}_{7}}\]
D.	None of these
Answer» C. \[^{14}{{C}_{7}}\]

Discussion

6163.	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
Answer» C. \[^{18}{{C}_{9}}\]

Discussion

6164.	\[^{n}{{C}_{r}}{{\div }^{n}}{{C}_{r-1}}=\] [MP PET 1984]
A.	\[\frac{n-r}{r}\]
B.	\[\frac{n+r-1}{r}\]
C.	\[\frac{n-r+1}{r}\]
D.	\[\frac{n-r-1}{r}\]
Answer» D. \[\frac{n-r-1}{r}\]

Discussion

6165.	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
Answer» C. 333

Discussion

6166.	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
Answer» B. 56

Discussion

6167.	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
Answer» C. 8150

Discussion

6168.	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
Answer» B. 840

Discussion

6169.	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
Answer» D. None of these

Discussion

6170.	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
Answer» D. None of these

Discussion

6171.	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
Answer» C. 455

Discussion

6172.	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
Answer» D. 80

Discussion

6173.	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
Answer» B. 60480

Discussion

6174.	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
Answer» B. \[^{22}{{C}_{9}}{{-}^{5}}{{C}_{3}}\]

Discussion

6175.	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
Answer» D. None of these

Discussion

6176.	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
Answer» C. 4340

Discussion

6177.	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
Answer» D. 42

Discussion

6178.	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
Answer» C. 75600

Discussion

6179.	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
Answer» B. \[\frac{52\ !}{{{(13\ !)}^{2}}\ 4\ !}\]

Discussion

6180.	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.	\[^{15}{{P}_{5}}\]
Answer» B. \[\frac{15\ !}{{{(3\ !)}^{5}}}\]

Discussion

6181.	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
Answer» B. 4

Discussion

6182.	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
Answer» B. \[^{104}{{C}_{26}}\]

Discussion

6183.	In a touring cricket team there are 16 players in all including 5 bowlers and 2 wicket-keepers. How many teams of 11 players from these, can be chosen, so as to include three bowlers and one wicket-keeper [MP PET 1984]
A.	650
B.	720
C.	750
D.	800
Answer» C. 750

Discussion

6184.	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
Answer» C. 62

Discussion

6185.	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.	\[^{n-p}{{C}_{r}}\ r\ !\]
C.	\[^{n-p}{{C}_{r-p}}\ r\ !\]
D.	None of these
Answer» D. None of these

Discussion

6186.	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
Answer» D. 892

Discussion

6187.	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
Answer» B. 5095

Discussion

6188.	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
Answer» C. 2008

Discussion

6189.	\[^{n}{{C}_{r}}{{+}^{n-1}}{{C}_{r}}+......{{+}^{r}}{{C}_{r}}\] = [AMU 2002]
A.	\[^{n+1}{{C}_{r}}\]
B.	\[^{n+1}{{C}_{r+1}}\]
C.	\[^{n+2}{{C}_{r}}\]
D.	\[{{2}^{n}}\]
Answer» C. \[^{n+2}{{C}_{r}}\]

Discussion

6190.	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
Answer» B. 1450

Discussion

6191.	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.	\[^{23}{{C}_{2}}\]
Answer» B. \[15+8\]

Discussion

6192.	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
Answer» D. 6

Discussion

6193.	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
Answer» D. None of these

Discussion

6194.	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
Answer» D. 120

Discussion

6195.	If \[^{43}{{C}_{r-6}}={{\,}^{43}}{{C}_{3r+1}},\] then the value of r is [Kerala (Engg.) 2002]
A.	12
B.	8
C.	6
D.	10
Answer» B. 8

Discussion

6196.	Value of r for which \[^{15}{{C}_{r+3}}={{\,}^{15}}{{C}_{2r-6}}\] is [Pb. CET 1999]
A.	2
B.	4
C.	6
D.	-9
Answer» D. -9

Discussion

6197.	If \[n\] and \[r\] are two positive integers such that \[n\ge r,\] then \[^{n}{{C}_{r-1}}\]\[+{{\,}^{n}}{{C}_{r}}=\] [Kerala (Engg.) 2002]
A.	\[^{n}{{C}_{n-r}}\]
B.	\[^{n}{{C}_{r}}\]
C.	\[^{n-1}{{C}_{r}}\]
D.	\[^{n+1}{{C}_{r}}\]
Answer» E.

Discussion

6198.	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
Answer» B. 10

Discussion

6199.	\[\left( \begin{matrix} n \\ n-r \\ \end{matrix} \right)\,+\,\left( \begin{matrix} n \\ r+1 \\ \end{matrix} \right)\], whenever \[0\le r\le n-1\]is equal to [AMU 2000]
A.	\[\left( \begin{matrix} n \\ r-1 \\ \end{matrix} \right)\]
B.	\[\left( \begin{matrix} n \\ r \\ \end{matrix} \right)\]
C.	\[\left( \begin{matrix} n \\ r+1 \\ \end{matrix} \right)\]
D.	\[\left( \begin{matrix} n+1 \\ r+1 \\ \end{matrix} \right)\]
Answer» E.

Discussion

6200.	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
Answer» B. \[n>7\]

Discussion

Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

If \[^{n}{{C}_{12}}={{\,}^{n}}{{C}_{6}}\], then \[^{n}{{C}_{2}}=\] [Karnataka CET 2005]

The value of \[{}^{50}{{C}_{4}}+\sum\limits_{r=1}^{6}{^{56-r}{{C}_{3}}}\] is [AIEEE 2005]

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]

The value of \[\sum\limits_{r=0}^{n-1}{\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r}}+{{\,}^{n}}{{C}_{r+1}}}}\] equals [MP PET 2004]

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 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]

If \[^{n}{{C}_{r}}\] denotes the number of combinations of n things taken r at a time, then the expression \[^{n}{{C}_{r+1}}+{{\,}^{n}}{{C}_{r-1}}+\,2\times {{\,}^{n}}{{C}_{r}}\] equals [AIEEE 2003]

A man has 10 friends. In how many ways he can invite one or more of them to a party [AMU 2002]

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]

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]

The number of ways in which four letters of the word 'MATHEMATICS' can be arranged is given by [Kurukshetra CEE 1996; Pb. CET 1995]

\[^{14}{{C}_{4}}+\sum\limits_{j=1}^{4}{^{18-j}{{C}_{3}}}\] is equal to [EAMCET 1991]

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]

\[^{n}{{C}_{r}}{{\div }^{n}}{{C}_{r-1}}=\] [MP PET 1984]

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

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

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

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

The number of ways in which any four letters can be selected from the word ?CORGOO? is

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

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

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]

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]

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]

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]

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]

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]

How many words of 4 consonants and 3 vowels can be formed from 6 consonants and 5 vowels [RPET 1985]

The number of ways of dividing 52 cards amongst four players equally, are [IIT 1979]

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]

If \[^{15}{{C}_{3r}}{{=}^{15}}{{C}_{r+3}}\], then the value of \[r\] is [IIT 1967; RPET 1991; MP PET 1998; Karnataka CET 1996]

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

In a touring cricket team there are 16 players in all including 5 bowlers and 2 wicket-keepers. How many teams of 11 players from these, can be chosen, so as to include three bowlers and one wicket-keeper [MP PET 1984]

Out of 6 books, in how many ways can a set of one or more books be chosen [MP PET 1984]

The numbers of permutations of \[n\] things taken \[r\] at a time, when \[p\]things are always included, is

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

In how many ways can a committee consisting of one or more members be formed out of 12 members of the Municipal Corporation

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

\[^{n}{{C}_{r}}{{+}^{n-1}}{{C}_{r}}+......{{+}^{r}}{{C}_{r}}\] = [AMU 2002]

In how many ways can 21 English and 19 Hindi books be placed in a row so that no two Hindi books are together

In how many ways can a girl and a boy be selected from a group of 15 boys and 8 girls

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

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

How many numbers of 6 digits can be formed from the digits of the number 112233 [Karnataka CET 2004]

If \[^{43}{{C}_{r-6}}={{\,}^{43}}{{C}_{3r+1}},\] then the value of r is [Kerala (Engg.) 2002]

Value of r for which \[^{15}{{C}_{r+3}}={{\,}^{15}}{{C}_{2r-6}}\] is [Pb. CET 1999]

If \[n\] and \[r\] are two positive integers such that \[n\ge r,\] then \[^{n}{{C}_{r-1}}\]\[+{{\,}^{n}}{{C}_{r}}=\] [Kerala (Engg.) 2002]

The least value of natural number n satisfying \[C(n,\,5)+C(n,\,6)\,\,>C(n+1,\,5)\] is [EAMCET 2002]

\[\left( \begin{matrix} n \\ n-r \\ \end{matrix} \right)\,+\,\left( \begin{matrix} n \\ r+1 \\ \end{matrix} \right)\], whenever \[0\le r\le n-1\]is equal to [AMU 2000]

If \[^{n}{{C}_{3}}+{{\,}^{n}}{{C}_{4}}>{{\,}^{n+1}}{{C}_{3}},\]then [RPET 1999]