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 chairs 1 to 4 and then men select from the remaining chairs. Find the total number of possible arrangements.
1440
If the letters of the word RACHIT are arranged in all possible ways as listed in dictionary order, then what is the rank of the word RACHIT?
481
A candidate is required to answer 7 questions out of 12 questions, which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. Find the number of different ways of doing the questions.
780
Out of 18 points in a plane, no three are collinear except five points which are collinear. Find the number of lines that can be formed joining the points.
144
We wish to select 6 persons from 8, but if the person A is chosen, then B must also be chosen. In how many ways can selections be made?
22
How many committees of five persons with a chairperson can be selected from 12 persons?
3960
How many automobile license plates can be made if each plate contains two different letters followed by three different digits?
46800
A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected from the lot.
200
Find the number of permutations of \(n\) distinct things taken \(r\) together in which three particular things must occur together.
\( \binom{n-3}{r-3} (r-2)! 3! \)
Find the number of different words that can be formed from the letters of the word TRIANGLE so that no vowels are together.
14400
Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is repeated.
112
There are 10 persons named \(P_1, P_2, \dots, P_{10}\). Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement \(P_1\) must occur whereas \(P_4\) and \(P_5\) do not occur. Find the number of such possible arrangements.
4200
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
1023
A box contains two white, three black, and four red balls. In how many ways can three balls be drawn if at least one black ball is included?
64
If \(^nC_{r-1} = 36\), \(^nC_r = 84\) and \(^nC_{r+1} = 126\), then find \(r\).
r = 3
Find the number of integers greater than 7000 that can be formed with digits 3, 5, 7, 8 and 9 where no digit is repeated.
192
If 20 lines are drawn in a plane such that no two are parallel and no three are concurrent, in how many points will they intersect?
190
In a certain city, telephone numbers have six digits, the first two digits being one of 41, 42, 46, 62 or 64. How many telephone numbers have all six digits distinct?
8400
In an examination, a student has to answer 4 questions out of 5, with questions 1 and 2 being compulsory. Determine the number of ways in which the student can make the choice.
3
A convex polygon has 44 diagonals. Find the number of its sides.
11
18 mice were placed in two experimental groups and one control group, with all groups equally large. In how many ways can the mice be placed into three groups?
The total number of mice is 18 and they must be divided into three equal groups of 6 each.
The number of ways to divide \(18\) distinct mice into groups of size \(6, 6, 6\) is given by:
\[ \dfrac{18!}{(6!)^3} \]
Thus, the required number of ways is \( \dfrac{18!}{(6!)^3} \).
A bag contains six white marbles and five red marbles. Find the number of ways in which four marbles can be drawn from the bag if:
(a) they can be of any colour
(b) two must be white and two red
(c) they must all be of the same colour
(a) Total marbles = 11. Number of ways to draw 4 of any colour:
\[ \binom{11}{4} \]
(b) Two white and two red:
\[ \binom{6}{2} \times \binom{5}{2} \]
(c) All marbles same colour:
All white: \( \binom{6}{4} \)
All red: \( \binom{5}{4} \)
Total: \( \binom{6}{4} + \binom{5}{4} \)
In how many ways can a football team of 11 players be selected from 16 players? How many of them will:
(i) include 2 particular players?
(ii) exclude 2 particular players?
Total ways to select 11 players from 16:
\[ \binom{16}{11} \]
(i) If 2 particular players are included, then we choose the remaining 9 from the remaining 14:
\[ \binom{14}{9} \]
(ii) If 2 particular players are excluded, then we select all 11 from the remaining 14:
\[ \binom{14}{11} \]
A sports team of 11 students is to be constituted, choosing at least 5 from Class XI and at least 5 from Class XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?
The team must contain:
Case 1: 5 from Class XI and 6 from Class XII
Case 2: 6 from Class XI and 5 from Class XII
Total ways:
\[ \binom{20}{5} \times \binom{20}{6} + \binom{20}{6} \times \binom{20}{5} \]
Since the two terms are identical, the total is:
\[ 2 \left( \binom{20}{5} \times \binom{20}{6} \right) \]
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has:
(i) no girls
(ii) at least one boy and one girl
(iii) at least three girls
(i) No girls means all 5 selected from 7 boys:
\[ \binom{7}{5} = 21 \]
(ii) At least one boy and one girl:
Total ways to choose 5 from 11:
\[ \binom{11}{5} \]
Subtract teams with all boys and all girls:
All boys: \( \binom{7}{5} \)
All girls: \( \binom{4}{5} = 0 \)
Thus, ways =
\[ \binom{11}{5} - \binom{7}{5} = 462 - 21 = 441 \]
(iii) At least 3 girls:
Possible combinations:
3 girls + 2 boys: \( \binom{4}{3} \binom{7}{2} \)
4 girls + 1 boy: \( \binom{4}{4} \binom{7}{1} \)
Total:
\[ \binom{4}{3} \binom{7}{2} + \binom{4}{4} \binom{7}{1} = 4 \times 21 + 1 \times 7 = 84 + 7 = 91 \]
If \(^{n}C_{12} = ^{n}C_{8}\), then \(n\) is equal to
20
12
6
30
The number of possible outcomes when a coin is tossed 6 times is
36
64
12
32
The number of different four digit numbers that can be formed with the digits 2, 3, 4, 7 using each digit only once is
120
96
24
100
The sum of the digits in unit place of all the numbers formed with the help of 3, 4, 5 and 6 taken all at a time is
432
108
36
18
Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to
60
120
7200
720
A five digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions. The total number of ways this can be done is
216
600
240
3125
Everybody in a room shakes hands with everybody else. The total number of handshakes is 66. The total number of persons in the room is
11
12
13
14
The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is
105
15
175
185
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is
6
18
12
9
The number of ways in which a team of eleven players can be selected from 22 players always including 2 of them and excluding 4 of them is
\(\binom{16}{11}\)
\(\binom{16}{5}\)
\(\binom{16}{9}\)
\(\binom{20}{9}\)
The number of 5-digit telephone numbers having at least one of their digits repeated is
90,000
10,000
30,240
69,760
The number of ways in which we can choose a committee from four men and six women so that the committee includes at least two men and exactly twice as many women as men is
94
126
128
None
The total number of 9-digit numbers which have all different digits is
10!
9!
9 \times 9!
10 \times 10!
The number of words which can be formed out of the letters of the word ARTICLE, so that vowels occupy the even place is
1440
144
7!
\(^{4}C_{4} \times ^{3}C_{3}\)
Given 5 different green dyes, four different blue dyes and three different red dyes, the number of combinations of dyes which can be chosen taking at least one green and one blue dye is
3600
3720
3800
3600
If \( ^nP_r = 840 \) and \( ^nC_r = 35 \), then \( r = \) ____.
7
\( ^{15}C_8 + ^{15}C_9 - ^{15}C_6 - ^{15}C_7 = \) ____.
0
The number of permutations of \( n \) different objects, taken \( r \) at a time, when repetitions are allowed, is ____.
n^r
The number of different words that can be formed from the letters of the word INTERMEDIATE such that no two vowels ever come together is ____.
151200
Three balls are drawn from a bag containing 5 red, 4 white and 3 black balls. The number of ways in which this can be done if at least 2 are red is ____.
80
The number of six-digit numbers, all digits of which are odd, is ____.
5^6
In a football championship, 153 matches were played. Every two teams played one match with each other. The number of teams participating in the championship is ____.
18
The total number of ways in which six '+' and four '−' signs can be arranged in a line such that no two '−' signs occur together is ____.
35
A committee of 6 is to be chosen from 10 men and 7 women so as to contain at least 3 men and 2 women. In how many different ways can this be done if two particular women refuse to serve on the same committee? ____.
7800
A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ____.
64
There are 12 points in a plane of which 5 points are collinear. The number of lines obtained by joining these points in pairs is \( {}^{12}C_2 - {}^{5}C_2 \).
False
Three letters can be posted in five letterboxes in \( 3^5 \) ways.
False
In the permutations of \( n \) things, \( r \) taken together, the number of permutations in which \( m \) particular things occur together is \( n^{m} P_{r-m} \times {}^{r}P_{m} \).
False
In a steamer there are stalls for 12 animals, and there are horses, cows and calves (not less than 12 each) ready to be shipped. They can be loaded in \( 3^{12} \) ways.
True
If some or all of \( n \) objects are taken at a time, the number of combinations is \( 2^n - 1 \).
True
There will be only 24 selections containing at least one red ball out of a bag containing 4 red and 5 black balls, given that balls of the same colour are identical.
True
Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. The number of ways in which the seating arrangements can be made is \( \dfrac{11!}{5!6!} (9!)(9!) \).
True
A candidate is required to answer 7 questions out of 12 questions divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. He can choose the seven questions in 650 ways.
False
To fill 12 vacancies there are 25 candidates of which 5 are from scheduled castes. If 3 vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of ways of selection is \( {}^{5}C_3 \times {}^{20}C_9 \).
False
Match the items in Column A with Column B.
| Column A | Column B |
|---|---|
(a) One book of each subject | (i) 3968 |
(b) At least one book of each subject | (ii) 60 |
(c) At least one book of English | (iii) 3255 |
| Column A | Matched Item from Column B |
|---|---|
(a) One book of each subject | (ii) 60 |
(b) At least one book of each subject | (iii) 3255 |
(c) At least one book of English | (i) 3968 |
Match the items in Column A with Column B.
| Column A | Column B |
|---|---|
(a) Boys and girls alternate | (i) 5! \u00d7 6! |
(b) No two girls sit together | (ii) 10! - 5! \u00d7 6! |
(c) All the girls sit together | (iii) (5!)^2 + (5!)^2 |
(d) All the girls are never together | (iv) 2! \u00b7 5! \u00b7 5! |
| Column A | Matched Item from Column B |
|---|---|
(a) Boys and girls alternate | (iii) (5!)^2 + (5!)^2 |
(b) No two girls sit together | (i) 5! \u00d7 6! |
(c) All the girls sit together | (iv) 2! \u00b7 5! \u00b7 5! |
(d) All the girls are never together | (ii) 10! - 5! \u00d7 6! |
Match the items in Column A with Column B.
| Column A | Column B |
|---|---|
(a) In how many ways committee can be formed | (i) 10C\u2082 \u00d7 20C\u2083 |
(b) In how many ways a particular professor is included | (ii) 10C\u2082 \u00d7 19C\u2082 |
(c) In how many ways a particular lecturer is included | (iii) 9C\u2081 \u00d7 20C\u2083 |
(d) In how many ways a particular lecturer is excluded | (iv) 10C\u2082 \u00d7 20C\u2083 |
| Column A | Matched Item from Column B |
|---|---|
(a) In how many ways committee can be formed | (iv) 10C\u2082 \u00d7 20C\u2083 |
(b) In how many ways a particular professor is included | (iii) 9C\u2081 \u00d7 20C\u2083 |
(c) In how many ways a particular lecturer is included | (ii) 10C\u2082 \u00d7 19C\u2082 |
(d) In how many ways a particular lecturer is excluded | (i) 10C\u2082 \u00d7 20C\u2083 |
Match the items in Column A with Column B.
| Column A | Column B |
|---|---|
(a) how many numbers are formed? | (i) 840 |
(b) how many numbers are exactly divisible by 2? | (ii) 200 |
(c) how many numbers are exactly divisible by 25? | (iii) 360 |
(d) how many of these are exactly divisible by 4? | (iv) 40 |
| Column A | Matched Item from Column B |
|---|---|
(a) how many numbers are formed? | (i) 840 |
(b) how many numbers are exactly divisible by 2? | (iii) 360 |
(c) how many numbers are exactly divisible by 25? | (iv) 40 |
(d) how many of these are exactly divisible by 4? | (ii) 200 |
Match the items in Column A with Column B.
| Column A | Column B |
|---|---|
(a) 4 letters are used at a time | (i) 720 |
(b) All letters are used at a time | (ii) 240 |
(c) All letters are used but the first is a vowel | (iii) 360 |
| Column A | Matched Item from Column B |
|---|---|
(a) 4 letters are used at a time | (iii) 360 |
(b) All letters are used at a time | (i) 720 |
(c) All letters are used but the first is a vowel | (ii) 240 |