III. INNOVACIONES DE LA COMPILACIÓN DE REFERENCIA 2013
3.2 Otras revisiones de fuentes de información y métodos
3.2.2 Innovaciones de metodologías de estimación
(i) Out of n non-concurrent and non-parallel straight lines points of intersection are = nC2
(ii) Out of ‘n’ points the number of straight lines are (when no three are collinear) = nC2
(iii) If out of n points m are collinear, then Number of straight lines = nC2 – mC2 + 1 (iv) To find number of diagonals
Number of diagonals = n(n – 3)/2 (v) Number of triangle formed from n points (when no three points are collinear)
(vi) Number of triangles out of n points in which m are collinear = nC3 – mC3
(vii) Number of triangles that can be formed out of n points (when none of the side is common to the sides of polygon)
= nC3 – nC1 – nC1. n-4C1
(viii) Number of parallelogram in two system of parallel lines (when I set contains m parallel lines and II set contains n parallel lines)
= nC2 × mC2
(ix) Number of squares m-1
= Σ (m – r) (n – r) ; (m < n) r=1
EXERCISE
1. The number of arrangements of letters of the word BANANA in which the two N’s do not appear adjointly is:
(a) 40
(b) 60
(c) 80
(d) 100
2. How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that odd digits occupy even positions?
(a) 16
(b) 36
(c) 60
(d) 180
3.
***Ten different letters of an alphabet are given. Words with 5 letters are formed from these given letters.Then the number of words which have atleast one letter repeated is:
(a) 69760
(b) 30240
(c) 99748
(d) None of these
4. How many numbers greater than 1000, but not greater than 4000 can be formed with the digits 0, 1, 2, 3, 4, repetition of digits being allowed?
(a) 374
(b) 375
(c) 376
(d) None of these
5.
The straight lines l1, l2, l3 are parallel and lie in the same plane. A total number of m points are taken on l1, n points on l2, k points on l3. the maximum number of triangle formed with vertices of these points are :(a)
m+n+kC3(b)
m+n+kC3 – mC3 – nC3 – kC3(c)
mC3+nC3+kC3(d) None of these
6. There are 10 lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is:
(a)
102(b) 1023
(c)
210(d) 10!
7. How many 10 digit numbers can be written by using the digits 1 and 2?
(a)
10C1+9C2(b)
210(c)
10C2(d) 10!
8.
Number of divisors of the form 4n + 2 (n ≥ 0) of the integer 240 is:(a) 4
(b) 8
(c) 10
(d) 3
9. A five-digit number divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition.
The total number of ways this can be done is:
(a) 216
(b) 240
(c) 600
(d) 3125
10. The number of positive integers which can be formed by using any number of digits from 0, 1, 2, 3, 4, 5 by using each digit not more than once in each number is:
(a) 1200
(b) 1500
(c) 1600
(d) 1630
11. The total number of 9 digit numbers which have all different digits is:
(a) 10!
(b) 9!
(c) 9.9!
(d) 10.10!
12. Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4, and then the men select the chairs from amongst the remaining. The number of possible arrangements is:
(a)
4C3× 4C2(b)
4C2 × 4P3(c)
4P2 × 4P3(d) None of these
13. From 4 officers and 8 Jawans, a committee of 6 is to be chosen to include exactly one officer. The number of such committees is:
(a) 160
(b) 200
(c) 224
(d) 300
14. Given 5 different green dyes, 4 different blue dyes and 3 different red dyes. The number of combinations of dyes which can be chosen taking at least one green and one blue dye is:
(a) 3600
(b) 3720
(c) 3800
(d) None of these
15. If a polygon has 44 diagonals, then the number of its sides are :
(a) 11
(b) 7
(c) 8
(d) None of these
16.
An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is:(a) 6
(b) 7
(c) 8
(d) 9
17. The number of five-digit telephone numbers having at least one of their digits repeated is:
(a) 90000
(b) 100000
(c) 30240
(d) 69760
18.
If n is an integer between 0 and 21, then the minimum value of n! (21-n)! is:(a) 9! 21!
(b) 10! 11!
(c) 20!
(d) 21!
19.
The total number of six digit numbers x1x2x3x4x5x6 having the property x1<x2≤x3<x4<x5≤x6, is equal to:(a)
10C6(b)
12C6(c)
11C6(d) None of these
20. The total number of three digit numbers, then sum of whose digits is even, is equal to:
(a) 450
(b) 350
(c) 250
(d) 325
21. If letters of the work ‘KUBER’ are written in all possible orders ad arranged as in a dictionary, then rank of the word ‘KUBER’ will be:
(a) 67
(b) 68
(c) 65
(d) 69
22. In a chess tournament, all participants were to play one game with the another. Two players fell ill after having played 3 games each. If total number of games played in the tournament is equal to 84, then total number of participants in the beginning was equal to:
(a) 10
(b) 15
(c) 12
(d) 14
23. In a country no two persons have identical set of teeth and there is no person with out a tooth, also no person has more than 32 teeth. If shape and size of tooth is disregarded and only the position of tooth is considered, then maximum population of that country can be:
(a)
232(b)
232-1(c) can’t be obtained (d) none of these
24. The total number of flags with three horizontal strips, in order, that can be formed using 2 identical red, 2 identical green and 2 identical white strips, is equal to:
(a) 4!
(b) 3.(4!) (c) 2.(4!) (d) None of these
25. The sides AB, BC, CA of a triangle ABC have 3, 4, 5 interior points respectively on them. Total number of triangles that can be formed using these points as vertices, is equal to:
(a) 135
(b) 145
(c) 178
(d) 205
26.
‘n’ different toys have to be distributed among ‘n’ children. Total number of ways in which these toys can be distributed so that exactly one child gets no toy, is equal to:(a)
n!(b)
n!nC2(c)
(n-1)!nC2(d)
n!n-1C227.
Total number of non-negative integral solutions of x1+x2+x3 = 10 is equal to:(a)
12C3(b)
10C3(c)
12C2(d)
10C228. A class contains 3 girls and four boys. Every Saturday five student go on a picnic, a different group of students is being sent each week. During the picnic, each girl in the group is given doll by the accompanying teacher. All possible groups of five have gone once, the total number of dolls the girls have got is:
(a) 21
(b) 45
(c) 27
(d) 24
29. total number of 4 digit number that are greater than 3000, that can be formed using the digits 1, 2, 3, 4, 5, 6 (no digit is being repeated in any number) is equal to:
(a) 120
(b) 240
(c) 480
(d) 80
30. A variable name in certain computer language must be either a alphabet or a alphabet followed by a decimal digit. Total number of different variable names that can exist in that language is equal to:
(a) 280
(b) 290
(c) 286
(d) 296
31. There are 10 person among whom two are brother. The total number of ways in which these persons can be seated around a round table so that exactly one person sit between the brothers, is equal to:
(a) (2!) (7!) (b) (2!) (8!) (c) (3!) (7!) (d) (3!) (8!)
PROBABILITY
Experiment
An operation which results in some well-defined outcomes is called an experiment.
Random Experiment
An experiment whose outcome cannot be predicted with certainty is called a random experiment. In other words, if an experiment is performed many times under similar conditions and the outcome of each time is not the same, then this experiment is called a random experiment.
Example: a) Tossing of a fair coin b) Throwing of an unbiased die
c) Drawing of a card from a well shuffled pack of 52 playing cards Sample Space
The set of all possible outcomes of a random experiments is called the sample space for that experiment. It is usually denoted by S.
Example:
f) When a die is thrown, any one of the numbers 1, 2, 3, 4, 5, 6 can come up. Therefore. Sample space
S = {1, 2, 3, 4, 5, 6}
g) When a coin is tossed either a head or tail will come up, then the sample space w.r.t. the tossing of the coin is
S = {H, T}
h) When two coins are tossed, then the sample space is Sample point / event point
Each element of the sample spaces is called a sample point or an event point.
Example: When a die is thrown, the sample space is S = {1, 2, 3, 4, 5, 6} where 1, 2, 3, 4, 5 and 6 are the sample points.
Discrete Sample Space
A sample space S is called a discrete sample if S is a finite set.
Event
A subset of the sample space is called an event.
Problem of Events
Sample space S plays the same role as universal set for all problems related to the particular experiment.
φ is also the subset of S and is an impossible Event. S is also a subset of S which is called a sure event or a certain event.
Types of Events