Sample QuestionsPermutation and Combinations questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
Total number of words formed by $2$ vowels and $3$ consonants taken from $4$ vowels and $5$ consonants is equal to.
Answer: C.
View full solution →The number of $5-$digit telephone numbers having atleast one of their digits repeated is.
- A
$90,000$
- B
$10,000$
- C
$30,240$
- ✓
$69,760$
Answer: D.
View full solution →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.
Answer: A.
View full solution →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.
Answer: D.
View full solution →The number of words which can be formed out of the letters of the word $\text{ARTICLE,}$ so that vowels occupy the even place is.
- A
$1440$
- ✓
$144$
- C
$7!$
- D
$\ ^4C_4 \times \ ^3C_3$
Answer: B.
View full solution →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
View full solution →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 other side of the table.
The number of ways in which the seating arrangements can be made is $\frac{11!}{5!6!}(9!)(9!) .$
View full solution →There are $12$ points in a plane of which $5$ points are collinear, then the number of lines obtained by joining these points in pairs is $^{12}C_2 – ^5C_2 .$
View full solution →To fill $12$ vacancies there are $25$ candidates of which$ 5$ are from scheduled castes. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all, the number of ways in which the selection can be made is $\ ^5C_3 \times \ ^{20}C_9 .$
In each if the Exercises from $60$ to$ 64$ match each item given under the column $C_1$ to its correct answer given under the column $C_2 $.
View full solution →Three letters can be posted in five letterboxes in $35$ ways.
View full solution →How many committee of five persons with a chairperson can be selected from $12$ persons.
$[$Hint: Chairman can be selected in $12$ ways and remaining in $11C_4 .]$
View full solution →A box contains two white, three black and four red balls. In how many ways can three balls be drawn from the box$,$ if atleast one black ball is to be included in the draw.
$[$Hint: Required number of ways $ =\ ^3C_1\times \ ^6C_2+ \ ^3C_2 \times \ ^6C_2 + \ ^3C_3 .]$
View full solution →Eight chairs are numbered $1$ to $8$. Two women and $3$ men wish to occupy one chair each. First the women choose the chairs from amongst the chairs $1$ to $4$ and then men select from the remaining chairs. Find the total number of possible arrangements.
$[$Hint: $2$ women occupy the chair$,$ from $1$ to $4$ in $^4P_2$ ways and $3$ men occupy the remaining chairs in $^6P_3$ ways.]
View full solution →If the letters of the word RACHIT are arranged in all possible ways as listed in dictionary. Then what is the rank of the word RACHIT?
[Hint: In each case number of words beginning with A, C, H, I is 5!]
In an examination$,$ a student has to answer $4$ questions out of $5$ questions; questions $1$ and $2$ are however compulsory. Determine the number of ways in which the student can make the choice.
View full solution →Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
How many words $($with or without dictionary meaning$)$ can be made from the letters of the word $\text{MONDAY,}$ assuming that no letter is repeated$,$ if
|
|
$C_1$
|
|
$C_2$
|
| $(a)$ |
$4$ 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.
|
$(iii)$ |
$360$ |
View full solution →Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
There are $3$ books on Mathematics$, 4$ on Physics and $5$ on English. How many different collections can be made such that each collection consists of :
|
|
$C_1$
|
|
$C_2$
|
| $(a)$ |
One book of each subject.
|
$(i)$ |
$3968$ |
| $(b)$ |
At least one book of each subject.
|
$(ii)$ |
$60$ |
| $(c)$ |
At least one book of English.
|
$(iii)$ |
$3255$ |
View full solution →There are $10$ persons named $P_1 , P_2 , P_3 , ... P_{10.}$ 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.
$[$Hint: Required number of arrangement $ = ^7C_4 \times 5!]$
View full solution →In a certain city, all telephone numbers have six digits, the first two digits always being $41$ or $42$ or $46$ or $62$ or $64$. How many telephone numbers have all six digits distinct?
View full solution →In how many ways can a football team of $11$ players be selected from $16$ players? How many of them will.
- Include $2$ particular players?
- Exclude $2$ particular players?
View full solution →The number of permutations of n different objects$,$ taken $r$ at a line$,$ when repetitions are allowed$,$ is $....$
View full solution →If $\ ^nP_r= 840, \ ^nC_r = 35,$ then $r =......$
View full solution →A committee of $6$ is to be chosen from $10$ men and $7$ women so as to contain atleast $3$ men and $2$ women. In how many different ways can this be done if two particular women refuse to serve on the same committee.
$[$Hint: At least $3$ men and $2$ women: The number of ways $= \ ^{10}C_3\times \ ^7C_3 + \ ^{10}C_4 \times \ ^7C_2 .$ For $2$ particular women to be always there: the number of ways $= \ ^{10}C4 + \ ^{10}C_3\times \ ^5C_1 $. The total number of committees when two particular women are never together$ \ =$ Total $–$ together$.]$
View full solution →$^{15}C_8 +^{15}C_9 – ^{15}C_6 – ^{15}C_7 = ......$
View full solution →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$.....$
View full solution →Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
Using the digits $1, 2, 3, 4, 5, 6, 7,$ a number of $4$ different digits is formed. Find
|
|
$C_1$
|
|
$C_2$ |
| $(a)$ |
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 divisble by $4.$
|
$(iv)$ |
$40$ |
View full solution →Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
Five boys and five girls form a line. Find the number of ways of making the seating arrangement under the following condition:
|
|
$C_1$
|
|
$C_2$ |
| $(a)$ |
Boys and girls alternate.
|
$(i)$ |
$5! \times 6!$ |
| $(b)$ |
No two girls sit together.
|
$(ii)$ |
$10! – 5! 6!$ |
| $(c)$ |
All the girls sit together.
|
$(iii)$ |
$(5!)^2+ (5!)^2$ |
| $(d)$ |
All the girls are never together.
|
$(iv)$ |
$2!\ 5!\ 5!$ |
View full solution →Match each item given under the column $C_1$ to its correct answer given under the column $C_2.$
There are $10$ professors and $20$ lecturers out of whom a committee of $2$ professors and $3$ lecturer is to be formed. Find:
|
|
$C_1$
|
|
$C_2$ |
| $(a)$ |
In how many ways committee can be formed.
|
$(i)$ |
$^{10}C_2 \times ^{19}C_3$ |
| $(b)$ |
In how many ways a particular professor is included.
|
$(ii)$ |
$^{10}C_2 \times ^{19}C_2$ |
| $(c)$ |
In how many ways a particular lecturer is included.
|
$(iii)$ |
$^9C_1 \times ^{20}C_3$ |
| $(d)$ |
In how many ways a particular lecturer is excluded.
|
$(iv)$ |
$^{10}C_2\times ^{20}C_3$ |
View full solution →Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.