Mathematics - Permutations and Combinations
Exam Duration: 45 Mins Total Questions : 30
A question paper is divided into two parts A and B. Each part contains 5 questions. The number of ways in which candidate can answer 6 questions selecting at least two questions from each part is
- (a)
50
- (b)
100
- (c)
200
- (d)
NONE OF THESE
m men and n women are to be seated in a row so that no two woman sit together. It m>n, then the number of ways in which this can be done is
- (a)
\(n!(m+1)!\over (m-n+1)!\)
- (b)
\(m!n!\over(m-n+1)!\)
- (c)
\(m!(m+1)!\over(m-n+1)!\)
- (d)
NONE OF THESE
Tn denotes the number of tiangles which can be formed using the verticles of a regular polygon of n sides. If Tn+1-Tn = 21, then n equals
- (a)
5
- (b)
7
- (c)
6
- (d)
NONE OF THESE
Number of divisors of the form \(4n+2(n\ge 0)\) of the integer 240, is
- (a)
4
- (b)
8
- (c)
10
- (d)
3
The number of words from the letters of the word 'BHARAT' in which B and H will never come together, is
- (a)
360
- (b)
240
- (c)
120
- (d)
None of the above
Observe the following columns:
| Column I | Column II |
|---|---|
| A. If \(\lambda\) be the number of ways in which 6 boys and 5 girls can be arranged in a line, so that they are alternate, then \(\lambda\) is divisible by |
p. 5! |
| B. I \(\lambda\) be the number of ways in which 6 boys and 5 girls can be seated in a row such that two girls are never together, then \(\lambda\) is divisible by |
q. 6! |
| C. If \(\lambda\) be the number of ways in which 6 boys and 5 girls can be seated around a round table if all the five girls do not sit together, then \(\lambda\) is divisible by |
r. 7! |
| s. 5!6! t. 5!7! |
- (a)
A B C (p,q,r) (p,q,s) (p,q,r,s) - (b)
A B C (p,q,r,s) (p,q,r,s,t) (p,q,r) - (c)
A B C (p,q,s) (p,q,r,s,t) (p,q,s) - (d)
None of the above
How many ways are there, to arrange the letters in the word 'GARDEN' with the vowels in alphabetical order?
- (a)
360
- (b)
240
- (c)
120
- (d)
480
The least positive integral value of X which satisfies the inequality 10Cx-1 > 2. 10Cx is:
- (a)
7
- (b)
8
- (c)
9
- (d)
10
The maximum number of points of intersection of 8 circles, is
- (a)
16
- (b)
24
- (c)
28
- (d)
56
The maximum number of points into which 4 circles and 4 straight lines intersect, is
- (a)
26
- (b)
50
- (c)
56
- (d)
72
The sum of the digits in the unit's place of all the numbers formed with the digits 5, 6,7,8 when taken all at a time, is
- (a)
104
- (b)
126
- (c)
127
- (d)
156
The number of triangles whose vertices arc the vertices of an octagon but none of whose sides happen to come from the octagon is
- (a)
16
- (b)
28
- (c)
56
- (d)
70
There are n points in a plane of which no three are in a straight line except 'm' which are all in a straight line. Then the number of different quadrilaterals, that can be formed with the given points as vertices. is
- (a)
nC4 - mC3n-m+1C1-mC4
- (b)
nC4 - mC3n-mC1+mC4
- (c)
nC4 - mC3n-mC1-mC4
- (d)
nC4 - nC3.mC1
The total numbers of seven digit numbers the sum of whose digits is even, is
- (a)
9 X 106
- (b)
45 X 105
- (c)
81 X 105
- (d)
9 X 105
Given that n is the odd, the number of ways in which three numbers in AP can be selected from 1, 2, 3, 4,..., n is
- (a)
\(\frac{(n-1)^{2}}{2}\)
- (b)
\(\frac{(n+1)^{2}}{4}\)
- (c)
\(\frac{(n+1)^{2}}{2}\)
- (d)
\(\frac{(n-1)^{2}}{4}\)
Eight straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. The number of parts into which these lines divide the plane, is
- (a)
29
- (b)
32
- (c)
36
- (d)
37
Seven different lecturers are to deliver lectures in seven periods of a class on a particular day. A, B and C are three of the lectures. The number of ways in which a routine for the day can be made such that A delivers his lecture before B, and B before C, is
- (a)
210
- (b)
420
- (c)
840
- (d)
none of these
In a city no 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 teeth and consider only the positioning of the teeth, then the maximum population of the city is
- (a)
232
- (b)
232 - 1
- (c)
232 - 2
- (d)
232 - 2
Sanjay has 10 friends among whom two are married to each other. She wishes to invite 5 of them for a party. If the married couple refuse to attend separately, then the number of different ways in which she can invite five friends is
- (a)
8C5
- (b)
2 X 8C3
- (c)
10C5 -2 X 8C4
- (d)
none of these
Suppose a lot of n objects contains n 1 objects of one kind, n 2 objects of second kind, n3 objects of third kind,....., n k objects of kth kind. Such that n1 + n2 + n3 + ... + n k = n, then the number of possible arrangements/permutations of r objects out of this lot is the coefficient of x r in the expansion of \(r!\Pi \left( \overset { { n }_{ 1 } }{ \underset { \lambda =0 }{ \Sigma } } \frac { { x }^{ \lambda } }{ \lambda ! } \right) \)
Five letter words are to be formed out of the letters of the word 'INFINITESIMAL', then number of permutations must be
- (a)
16995
- (b)
5665
- (c)
11330
- (d)
22660
Let f(n)f(n) denotes the number of different ways the positive integer n can be expressed as the sum of 1's and 2's. For example
f(4)=5f(4)=5
ie, 4=1+1+1+1
=1+1+2
=1+2+1
=2+1+1
=2+2
The number of solutions of the equation f(n) = n + 1, where n \(\in\) N is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
Different words are being formed by arranging the letters of the word 'ARRANGE'. All the words obtained are written in the form of a dictionary.
The number of words in which neither two 'R' nor two 'A' come together is
- (a)
1260
- (b)
660
- (c)
900
- (d)
240
Five digit number divisible by 3 is formed using 0,1,3,4,6,7 without repetition. Total number of such numbers are
- (a)
312
- (b)
3125
- (c)
120
- (d)
7216
Find the number of different 8 letter arrangements that can be made from the letters of the world DAUGHTER so that
(i) all vowels occur together
(ii) all vowels do not occur together
- (a)
4320,36000
- (b)
4300,36000
- (c)
4200,36000
- (d)
4300,3600
In how many ways can 4 red, 3yellow and 2 green discs be arranged in a row of the discs of the same color are indistinguishable?
- (a)
1200
- (b)
1220
- (c)
1240
- (d)
1260
Find the number of arrangements of the letters of the world INDEPENDENCE. In how many of these arrangements, do the word with P?
- (a)
1663200,138600
- (b)
163000,135600
- (c)
160000,138650
- (d)
None of these
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)
52C18X35C2
- (b)
52C18X35C2+52C19X35C1+52C20
- (c)
52C18+35C2+52C19
- (d)
52C18X35C2+35C1X52C19
In how many ways a committee consiting of 3 men and 2 women, can be chosen from 7 men and 5 women?
- (a)
45
- (b)
350
- (c)
4200
- (d)
230
There were two women participants in a chess tournament. The number of games the men played between themselves exceeded by 52 the number of games they played with women. If each player played one game with each other, the number of men in the tournament ,was
- (a)
10
- (b)
11
- (c)
12
- (d)
13
1654060945.png)
1654060942.png)
1653987551.png)
1653987086.png)
1582095229.jpg)
