Combination of n Different Things Taken One or More at a Time

Q. A woman has $11$ close friends. The number of ways in which she can invite $5$ of them to dinner, if two particular of them are not on speaking terms and will not attend together, is

1. ${}^{11}{C}_5-{}^9{C}_3$
2. ${}^9{C}_5+2\times {}^9{C}_4$
3. $3\times {}^9{C}_4$
4. ${}^9{C}_4$

Q. All the arrangements, with or without meaning, of the word $FARMER$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $FARMER$ in this list is

Q.

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE ?

Q. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

1. 150
2. 75
3. 210
4. 243

Q.

From 4 officers and 8 jawans in how many ways can 6 be chosen (i) to include exactly one officer (ii) to include at least one officer ?

Q. Find the number of different signals that can be generated by arranging at least $2$ flags in order (one below the other) on a vertical staff, if five different flags are available?

Q. A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750$, then $n$ is equal to :

1. $24$
2. $27$
3. $28$
4. $25$

Q.

We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can the selection be made ?

Q. Poor Dolly’s T.V. has only $4$ channels, all of them quite boring, hence it is not surprising that she desires to switch (change) channel after every one minute. Find the number of ways in which she can change the channels so that she is back to her original channel for the first time after $4$ minutes.

1. $6$
2. $3$
3. $12$
4. $24$

Q.

From a class of 12 boys and 10 girls, 10 students are to be chosen for a competition; at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made ?

Q. A man has $3$ friends. The number of ways he can invite one friend everyday for dinner on $6$ successive nights so that no friend is invited more than $3$ times is $N$, then $N/170$

Q. A house of height $100$ metres subtends a right angle at the window of an opposite house. If the height of the window is $64$ metres, then the distance between the two houses is

1. $64$
2. $48$
3. $16$
4. $32$

Q.

Find the number of ways in which :

(a) a selection (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'.

Q. The number of ways in which the letters of the word ARRANGE can be permuted such that the R’s occur together is

1. $\frac{7!}{2!2!}$
2. $\frac{7!}{2!}$
3. $\frac{6!}{2!}$
4. $5!\times 2!$

Q. If the number of ways of selecting $3$ numbers out of $1,2,3,\dots ,2n+1$ such that they form an increasing arithmetic progression is $441,$ then the sum of the divisors of $n$ is equal to

1. $21$
2. $32$
3. $45$
4. $60$

Q. Let $N=10800.$ Then the correct option(s) among the following is/are

1. Number of even divisors is $48$
2. Number of odd divisors is $12$
3. Number of divisors that are multiple of $3$ is $45$
4. Number of divisors that are multiple of $15$ is $24$

Q.

The number of ways in which five identical balls can be distributed among ten identical boxes such that no box contains more than one ball, is?

1. $10!$

2. $\frac{10!}{5!}$

3. $\frac{10!}{{\left(5!\right)}^2}$

4. None of these

Q. Let $N=2^6{.5}^5{.7}^6{.10}^7$, then the total number of even factors of $N$ is

1. $1183$
2. $1200$
3. $1050$
4. $540$

Q. The sum of all natural numbers less than 200, that are divisible neither by 3 nor by 5, is

1. 10730
2. 10732
3. 15375
4. None of these

Q. The sum of the digit in the unit’s place of four digit numbers formed using digits $3,4,5,6$ taken all at a time is

Q. If the number of arrangements of letters of the word $DHARAMSHALA$ taken all at a time so that no two alike letters appear together is $4^a\cdot 5^b\cdot 6^c\cdot 7^d,$ where $a,b,c,d\in N,$ then the value of $a+b+c+d$ is

Q. A class contains three girls and four boys. Every Saturday, a group of $5$ students go on a picnic (a different group of students is sent every week). During the picnic, each girl in the group is given a doll by the accompanying teacher. If all possible groups of five have gone for picnic once, the total number of dolls that the girls have got is

1. $21$
2. $45$
3. $27$
4. $24$

Q.

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempts 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions ?

Q.

Three persons enter a railway compartment. If there are 5 seats a vacant, in how many ways can they take these seats ?

1. 20

2. 15

3. 125

4. 60

Q. The number of ways of selecting $15$ teams from $15$ men and $15$ women, such that each team consists of a man and a woman, is :

1. $({{}^{15}{C}_1)}^2+{{(}^{14}{C}_1)}^2+....+({{}^1{C}_1)}^2$
2. $15!$
3. $15{(}^{15}{C}_1{\times }^{15}{C}_1)$
4. ${}^{15}{C}_1\times 15!$

Q.

How many numbers of six digits can be formed from the digits 0, 1, 3, 5, 7 and 9 when no digit is repeated? How many of them are divisible by 10?

Q. The number of ways in which one or more balls can be selected out of $10$ white, $9$ green and $7$ black balls is

1. $892$
2. $881$
3. $891$
4. $879$

Q. In how many ways can $17$ persons depart from railway station in $2$ cars and $3$ autos, given that $2$ particular persons depart by same car ($4$ persons can sit in a car and $3$ persons can sit in an auto)?

1. $\frac{15!}{2!4!(3!{)}^3}$
2. $\frac{16!}{(2!{)}^{2}4!(3!{)}^3}$
3. $\frac{17!}{2!4!(3!{)}^3}$
4. $\frac{15!}{4!(3!{)}^3}$

Q.

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 teams be constituted ?

Q. A palindrome is a word, number, phrase or sequence of words that reads the same backwards as forwards e.g. "SOLOS".
The number of palindromes that can be formed using the letters ''AABBBBCCCDDDD'' is

1. $\frac{6!}{2!}$
2. $\frac{6!}{2!\times 2!}$
3. $\frac{7!}{2!\times 2!}$
4. $\frac{8!}{4!\times 4!}$