
Three boys and three girls are to be seated around a table, in a circle. Among them, the boy X does not want any girl neighbour and the girls Y does not want any boy neighbour. The number of such arrangements possible is
A)
4 done
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B)
6 done
clear
C)
8 done
clear
D)
None of these done
clear
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There are three men and seven women taking a dance class. Number of different ways in which each man is paired with a woman partner, and the four remaining women are paired into two pairs each of two is
A)
105 done
clear
B)
315 done
clear
C)
630 done
clear
D)
450 done
clear
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Seven people leave their bags outside temple and while returning after worshiping the deity, picked one bag each at random. In how many ways at least one and at most three of them get their correct bags?
A)
\[^{7}{{C}_{3}}.9{{+}^{7}}{{C}_{5}}.44{{+}^{7}}{{C}_{1}}.265\] done
clear
B)
\[^{7}{{C}_{6}}.265{{+}^{7}}{{C}_{5}}.9{{+}^{7}}{{C}_{7}}.44\] done
clear
C)
\[^{7}{{C}_{5}}.9{{+}^{7}}{{C}_{2}}.44{{+}^{7}}{{C}_{1}}.265\] done
clear
D)
None of these done
clear
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Four couples (husband and wife) decide to form a committee of four members. Find the number of different committees that can be formed in which no couple finds a place.
A)
12 done
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B)
14 done
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C)
16 done
clear
D)
24 done
clear
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The sides AB, BC, CA, of a triangle ABC have 3, 4 and 5 interior points respectively on them. The total number of triangles that can be constructed by using these points as vertices is
A)
220 done
clear
B)
204 done
clear
C)
205 done
clear
D)
195 done
clear
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A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is:
A)
40 done
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B)
41 done
clear
C)
16 done
clear
D)
32 done
clear
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If all permutations of the letters of the word AGAIN are arranged as in dictionary, then fiftieth word is
A)
NAAGI done
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B)
NAGAI done
clear
C)
NAAIG done
clear
D)
NAIAG done
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How many numbers with no more than three digits can be formed using only the digits 1 through 7 with no digit used more than once in a given number?
A)
259 done
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B)
249 done
clear
C)
257 done
clear
D)
252 done
clear
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The value of \[{{1}^{2}}.{{C}_{1}}+{{3}^{2}}.{{C}_{3}}+{{5}^{2}}.{{C}_{5}}+...\] is:
A)
\[n{{(n1)}^{n2+n{{.2}^{n1}}}}\] done
clear
B)
\[n{{(n1)}^{n2}}\] done
clear
C)
\[n{{(n1)}^{n3}}\] done
clear
D)
None of these done
clear
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In how many ways vertices of a square can be coloured with 4 distinct colour if rotations are considered to be equivalent, but reflections are distinct?
A)
65 done
clear
B)
70 done
clear
C)
71 done
clear
D)
None of these done
clear
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The total number of 5 digit numbers of different digits in which the digit in the middle is the largest, is
A)
\[\sum\limits_{n=4}^{9}{^{n}{{P}_{4}}}\] done
clear
B)
\[\sum\limits_{n=4}^{9}{^{n}{{P}_{4}}}\frac{1}{3!}\sum\limits_{n=3}^{9}{^{n}{{P}_{3}}}\] done
clear
C)
\[30(3!)\] done
clear
D)
None of these done
clear
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If\[n={{2}^{p1}}({{2}^{p}}1)\], where \[{{2}^{p}}1\] is a prime, then the sum of the divisors of n is equal to
A)
n done
clear
B)
2n done
clear
C)
pn done
clear
D)
\[{{p}^{n}}\] done
clear
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The value of ?n? for which \[^{n1}{{C}_{4}}{{}^{n1}}{{C}_{3}}\frac{5}{4}{{.}^{n1}}{{P}_{2}}<0,\] Where \[n\in N\]
A)
\[\{5,6,7,8,9,10\}\] done
clear
B)
\[\{1,2,3,4,5,6,7,8,9,10\}\] done
clear
C)
\[\{1,4,5,6,7,8,9,10\}\] done
clear
D)
\[(\infty ,2)\cup (3,11)\] done
clear
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Let P be a prime number such that \[p\ge 11.\] Let \[n=p!+1.\] The number of primes in the list \[n+1,\] \[n+2,n+3,...n+P1,\] is
A)
\[p1\] done
clear
B)
2 done
clear
C)
1 done
clear
D)
None of these done
clear
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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
A)
346 done
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B)
140 done
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C)
196 done
clear
D)
280 done
clear
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If \[S=(1)(1!)+(2)(2!)+(3)(3!)+...+n(n!),\] then
A)
\[\frac{S+1}{n!}\in \]integer done
clear
B)
\[\frac{S+1}{n!}\notin \] Integer done
clear
C)
\[\frac{S+1}{n!}\] cannot be discussed done
clear
D)
None of these done
clear
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Find the number of integral solution of the equation \[x+y+z=20\] and \[x>1,y>2\] and \[z>3.\]
A)
\[^{25}{{C}_{23}}\] done
clear
B)
\[^{17}{{C}_{2}}\] done
clear
C)
\[^{23}{{C}_{2}}\] done
clear
D)
None of these done
clear
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There are 10 bags \[{{B}_{1}},{{B}_{2}},{{B}_{3}},...,{{B}_{10}},\]which contain 21, 22? 30 different articles respectively. The total number of ways to bring out 10 articles from a bag is
A)
\[^{31}{{C}_{20}}{{}^{21}}{{C}_{10}}\] done
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B)
\[^{31}{{C}_{21}}\] done
clear
C)
\[^{31}{{C}_{20}}\] done
clear
D)
None of these done
clear
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If 16 identical pencils are distributed among 4 children such that each gets at least 3 pencils. The number of ways of distributing the pencils is
A)
15 done
clear
B)
25 done
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C)
35 done
clear
D)
40 done
clear
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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 done
clear
B)
2002 done
clear
C)
2008 done
clear
D)
8002 done
clear
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The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is
A)
\[^{8}{{C}_{3}}\] done
clear
B)
21 done
clear
C)
\[{{3}^{8}}\] done
clear
D)
5 done
clear
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The number of six digit numbers that can be formed from the digits \[1,2,3,4,5,6\] and 7 so that digits do not repeat and the terminal digits are even, is
A)
144 done
clear
B)
72 done
clear
C)
288 done
clear
D)
720 done
clear
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There are 10 points in a plane, no three are collinear, except 4 which are collinear. All points are joined. Let L be the number of different straight lines and T be the number of different triangles, then
A)
\[T=120\] done
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B)
\[L=40\] done
clear
C)
\[T=3L5\] done
clear
D)
None of these done
clear
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In a 12 storey house ten people enter a lift cabin. It is known that they will left in groups of 2,3 and 5 people at different storeys. The number of ways they can do so if the left does not stop to the second storey is
A)
78 done
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B)
112 done
clear
C)
720 done
clear
D)
132 done
clear
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A person writes letters to six friends and be the number of ways so that at least two of the number of ways so that all the letters are in wrong envelopes. Then \[\operatorname{x}  y =\]
A)
719 done
clear
B)
265 done
clear
C)
454 done
clear
D)
None done
clear
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A person invites a party of 10 friends at dinner and place them so that 4 are on one round table and 6 on the other round table. The number of ways in which he can arrange the guests is
A)
\[\frac{(10)!}{6!}\] done
clear
B)
\[\frac{(10)!}{24}\] done
clear
C)
\[\frac{(9)!}{24}\] done
clear
D)
None of these done
clear
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Ten different letters of an alphabet are given, words with five letters are formed form these given letters. Then the number of words which have at least one letter repeated is
A)
69760 done
clear
B)
30240 done
clear
C)
99784 done
clear
D)
None of these done
clear
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A boat is to be manned by eight men of whom 2 can only row on bow side and 3 can only row on stroke side, the number of ways in which the crew can be arranged is
A)
4360 done
clear
B)
5760 done
clear
C)
5930 done
clear
D)
None of these done
clear
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The number of distinct rational numbers x such that \[0<x<1\] and \[x=\frac{p}{q},\] where \[p,q\in \{1,2,3,4,5,6\},\] is
A)
15 done
clear
B)
13 done
clear
C)
12 done
clear
D)
11 done
clear
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In a small village, there are 87 families, of which 52 families have at most 2 children. In a rural development programme 20 families are to be chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made?
A)
\[^{52}{{C}_{18}}^{35}{{C}_{2}}\] done
clear
B)
\[^{52}{{C}_{18}}{{\times }^{35}}{{C}_{2}}{{+}^{52}}{{C}_{19}}{{\times }^{35}}{{C}_{1}}{{+}^{52}}{{C}_{20}}\] done
clear
C)
\[^{52}{{C}_{18}}{{+}^{35}}{{C}_{2}}{{+}^{52}}{{C}_{19}}\] done
clear
D)
\[^{52}{{C}_{18}}{{\times }^{35}}{{C}_{2}}{{+}^{35}}{{C}_{1}}{{\times }^{52}}{{C}_{19}}\] done
clear
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The straight lines \[{{\ell }_{1}},{{\ell }_{2}},{{\ell }_{3}}\] and parallel and lie in the same plane. A total number of m points are taken on \[{{\ell }_{1}}\], n points on \[{{\ell }_{2}}\]. k points on \[{{\ell }_{3}}\]. The maximum number of triangles formed with vertices at these points are:
A)
\[^{m+n+k}{{C}_{3}}\] done
clear
B)
\[^{m+n+k}{{C}_{3}}{{}^{m}}{{C}_{3}}{{}^{n}}{{C}_{3}}{{}^{k}}{{C}_{3}}\] done
clear
C)
\[^{m}{{C}_{3}}{{+}^{m}}{{C}_{3}}{{+}^{k}}{{C}_{3}}\] done
clear
D)
None of these done
clear
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All the words that can be formed using alphabets A, H, L, U and R are written as in a dictionary (no alphabet is repeated). Rank of the word RAHUL is
A)
71 done
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B)
72 done
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C)
73 done
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D)
74 done
clear
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'n' is selected form the set \[\{1,2,3...,100\}\]and the number \[{{2}^{n}}+{{3}^{n}}+{{5}^{n}}\] is formed. Total number of ways of selecting 'n' so that the formed number is divisible by 4, is equal to
A)
50 done
clear
B)
49 done
clear
C)
48 done
clear
D)
None of these done
clear
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Let \[1\le m<n\le p.\]The number of subsets of the set \[A=\{1,2,3,...p\}\] having m, n as the least and the greatest elements respectively, is
A)
\[{{2}^{nm1}}1\] done
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B)
\[{{2}^{nm1}}\] done
clear
C)
\[{{2}^{nm}}\] done
clear
D)
\[{{2}^{pn+m1}}\] done
clear
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The number of ways in which a mixed doubles game in tennis can be arranged form 5 married couples, if no husband and wife play in the same game, is
A)
46 done
clear
B)
54 done
clear
C)
60 done
clear
D)
None of these done
clear
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The total number of 3digit numbers, the sum of whose digits is even, is equal to
A)
450 done
clear
B)
350 done
clear
C)
250 done
clear
D)
325 done
clear
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There are 18 points in a plane such that no three of them are in the same line except five points which are collinear. The number of triangles formed by these points is:
A)
816 done
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B)
806 done
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C)
805 done
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D)
813 done
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Find the number of nonnegative solutions of the systems of equations: \[a+b=10,a+b+c+d=21,a+b+c+d+e+f=33,\]\[a+b+c+d+e+f+g+h=46\], and so on till \[a+b+c+d+....+x+y+z=208.\]
A)
\[^{22}{{P}_{10}}\] done
clear
B)
\[^{22}{{P}_{11}}\] done
clear
C)
\[^{22}{{P}_{13}}\] done
clear
D)
None of these done
clear
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Two straight line intersect at a point O. Points \[{{A}_{1}},{{A}_{2}},...{{A}_{n}}\] are taken on one line and pints \[{{B}_{1}},{{B}_{2}},...,{{B}_{n}}\]on the other. If the point O is not to be used, the number of triangles that can be drawn using these points as vertices, is
A)
\[n(n1)\] done
clear
B)
\[n{{(n1)}^{2}}\] done
clear
C)
\[{{n}^{2}}(n1)\] done
clear
D)
\[{{n}^{2}}{{(n1)}^{2}}\] done
clear
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5  Digit numbers are to be formed using 2, 3, 5, 7, 9 without repeating the digits. If p be the number of such numbers that exceed 20000 and q be the number of those that lie between 30000 and 90000, then p:q is:
A)
\[6:5\] done
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B)
\[3:2\] done
clear
C)
\[4:3\] done
clear
D)
\[5:3\] done
clear
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Number of ways in which 20 different pearls of two colours can be set alternately on a necklace, there being 10 pearls of each colour.
A)
\[6\times {{(9!)}^{2}}\] done
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B)
12! done
clear
C)
\[4\times {{(8!)}^{2}}\] done
clear
D)
\[5\times {{(9!)}^{2}}\] done
clear
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If 12 persons are seated in a row, the number of ways of selecting 3 persons from them, so that no two of them are seated next to each other is
A)
85 done
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B)
100 done
clear
C)
120 done
clear
D)
240 done
clear
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The number of permutations of the letters of the word HINDUSTAN such that neither the pattern ?HIN? nor ?DUS? nor ?TAN? appears, are
A)
166674 done
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B)
169194 done
clear
C)
166680 done
clear
D)
181434 done
clear
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A shop keeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in this stock. There are 5 places in a row in his show case. The number of different ways of displaying the three varieties of perfumes in the show case is
A)
6 done
clear
B)
50 done
clear
C)
150 done
clear
D)
None of these done
clear
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Consider n points in a plane no three of which are collinear and the ratio of number of hexagon and octagon that can be formed from these n points is 4:13 then find the value of n.
A)
14 done
clear
B)
20 done
clear
C)
28 done
clear
D)
None of these done
clear
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A seven digit number divisible by 9 is to be formed by using 7 out of number\[\{1,2,3,4,5,6,7,8,9\}\]. The number of ways in which this can be done is
A)
\[7!\] done
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B)
\[2\times 7!\] done
clear
C)
\[3\times 7!\] done
clear
D)
\[4\times 7!\] done
clear
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Total number of words that can be formed using all letters of the word FAILIRE which neither begin with F nor end with E is equal to
A)
3720 done
clear
B)
5040 done
clear
C)
3600 done
clear
D)
3480 done
clear
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The expression \[^{n}{{C}_{r}}+{{4.}^{n}}{{C}_{r1}}+{{6.}^{n}}{{C}_{r2}}+{{4.}^{n}}{{C}_{r3}}\]\[{{+}^{n}}{{C}_{r4}}\] is equal to
A)
\[^{n+4}{{C}_{r}}\] done
clear
B)
\[{{2.}^{n+4}}{{C}_{r1}}\] done
clear
C)
\[{{4.}^{n}}{{C}_{r}}\] done
clear
D)
\[{{11.}^{n}}{{C}_{r}}\] done
clear
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A teaparty is arranged for 16 people along two sides of a large table with 8 chairs on each side. Four men want to sit on one particular side and two on the other side. The number of ways in which they can be seated is
A)
\[\frac{6!8!10!}{4!6!}\] done
clear
B)
\[\frac{8!8!10!}{4!6!}\] done
clear
C)
\[\frac{8!8!6!}{6!4!}\] done
clear
D)
None of these done
clear
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The greatest common divisor of \[^{20}{{C}_{1}}{{,}^{20}}{{C}_{3}},...{{,}^{20}}{{C}_{19}}\] is
A)
20 done
clear
B)
4 done
clear
C)
5 done
clear
D)
None of these done
clear
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Six teachers and sic students have to sit round a circular table such that there is a teacher between any two students. The number of ways I which they can sit is
A)
\[6!\times 6!\] done
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B)
\[5!\times 6!\] done
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C)
\[5!\,\times 5!\] done
clear
D)
None of these done
clear
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Let \[E=(2n+1)(2n+3)(2n+5)...(4n3)(4n1);\] \[n>1\] then \[{{2}^{n}}E\] divisible by
A)
\[^{n}{{C}_{n/2}}\] done
clear
B)
\[^{2n}{{C}_{n}}\] done
clear
C)
\[^{3n}{{C}_{n}}\] done
clear
D)
\[^{4n}{{C}_{2n}}\] done
clear
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In a plane there are 37 straight lines of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no line passes through both points A and B, and no two are parallel. Then the number of intersection points the lines have is equal to
A)
535 done
clear
B)
601 done
clear
C)
728 done
clear
D)
None of these done
clear
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The number of different words which can be formed from the letters of the word LUCKNOW when the vowels always occupy even places in
A)
120 done
clear
B)
720 done
clear
C)
400 done
clear
D)
None of these done
clear
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The least positive integral values of n which satisfies the inequality \[^{10}{{C}_{n1}}>{{2.}^{10}}{{C}_{n}}\]
A)
7 done
clear
B)
8 done
clear
C)
9 done
clear
D)
10 done
clear
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If a denotes the number of permutations of x+2 things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of x 11 things taken all at a time such that \[a=182\,\,bc,\] then the value of x will be
A)
12 done
clear
B)
10 done
clear
C)
8 done
clear
D)
6 done
clear
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If \[\frac{2}{9!}+\frac{2}{3!7!}+\frac{1}{5!5!}=\frac{{{2}^{a}}}{b!},\] where \[a,b\in N,\]then the ordered pair (a, b) is
A)
\[(9,10)\] done
clear
B)
\[(10,9)\] done
clear
C)
\[(7,10)\] done
clear
D)
\[(10,7)\] done
clear
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The set \[S=\{1,2,3,...,12\}\] is to be partitioned into three sets, A, B, C of equal size. Thus\[A\cup B\cup C=S,A\cap B=B\cap C=A\cap C=\phi \]. The number of ways to partition S is
A)
\[\frac{12!}{{{(4!)}^{3}}}\] done
clear
B)
\[\frac{12!}{{{(4!)}^{4}}}\] done
clear
C)
\[\frac{12!}{3!{{(4!)}^{3}}}\] done
clear
D)
\[\frac{12!}{3!{{(4!)}^{4}}}\] done
clear
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Ravish writes letters to his five friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes?
A)
109 done
clear
B)
118 done
clear
C)
119 done
clear
D)
None of these done
clear
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\[f:\{1,2,3,4,5\}\to \{1,2,3,4,5\}\] that are onto and \[f(i)\ne i,\] is equal to
A)
9 done
clear
B)
44 done
clear
C)
16 done
clear
D)
None of these done
clear
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Let \[{{T}_{n}}\] denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If \[{{T}_{n+1}}{{T}_{n}}=21\], then n equals
A)
5 done
clear
B)
7 done
clear
C)
6 done
clear
D)
4 done
clear
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How many different nine digit numbers can be formed form the number 223355888 by rearranging its digits so that the odd digits occupy even positions?
A)
16 done
clear
B)
36 done
clear
C)
60 done
clear
D)
180 done
clear
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Let \[S=\sum\limits_{k=0}^{n1}{^{k+2}{{P}_{2}},}\] then
A)
n divides 3S done
clear
B)
n+1 divides 3S done
clear
C)
n+2 divides 3S done
clear
D)
All are correct done
clear
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How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?
A)
18 done
clear
B)
28 done
clear
C)
6 done
clear
D)
27 done
clear
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How many different words can be formed by jumbling the letters in the word MISSISSIOOI in which no two S are adjacent?
A)
\[{{8.}^{6}}{{C}_{4}}{{.}^{7}}{{C}_{4}}\] done
clear
B)
\[{{6.7.}^{8}}{{C}_{4}}\] done
clear
C)
\[{{6.8.}^{7}}{{C}_{4}}.\] done
clear
D)
\[{{7.}^{6}}{{C}_{4}}{{.}^{8}}{{C}_{4}}\] done
clear
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From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is:
A)
At least 500 but less than 750 done
clear
B)
At least 750 but less than 1000 done
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C)
At least 1000 done
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D)
Less than 500 done
clear
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The sum of all the numbers of four different digits that can be made by using the digits 0, 1, 2 and 3.
A)
64322 done
clear
B)
48522 done
clear
C)
38664 done
clear
D)
1000 done
clear
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The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is:
A)
\[^{30}{{C}_{7}}\] done
clear
B)
\[^{21}{{C}_{8}}\] done
clear
C)
\[^{21}{{C}_{7}}\] done
clear
D)
\[^{30}{{C}_{8}}\] done
clear
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Let \[A=\{xx\] is prime number and\[x<30\}\]. The number of different rational numbers whose numerator and denominator belong to A is
A)
90 done
clear
B)
180 done
clear
C)
91 done
clear
D)
100 done
clear
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If \[^{n}{{C}_{r1}}{{+}^{n+1}}{{C}_{r1}}{{+}^{n+2}}{{C}_{r1}}+...{{+}^{2n}}{{C}_{r1}}\] \[{{=}^{2n+1}}{{C}_{{{r}^{2}}132}}{{}^{n}}{{C}_{r}},\] Then the value of r and the minimum value of n are
A)
10 done
clear
B)
11 done
clear
C)
12 done
clear
D)
13 done
clear
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