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

Q. Sum of proper divisors of the number $1260$ is

Q. Which of the following has most number of divisors?

1. $99$
2. $101$
3. $176$
4. $182$

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. $25$
3. $27$
4. $28$

Q. If $P=2^4\cdot 6^3\cdot 5^2\cdot {15}^2$, then the number of proper even divisors of $P$ will be

1. $119$
2. $120$
3. $209$
4. $210$

Q. A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each X and Y are in this party, is:

1. $485$
2. $468$
3. $469$
4. $484$

Q. The number of proper divisors of $2160$ is

1. $40$
2. $39$
3. $38$
4. $18$

Q. Let S = {1, 2, 3, ………, 9}. For k = 1, 2, ……..5, let $N_k$ be the number of subsets of S, each containing five elements out of which exactly k are odd. Then $N_1+N_2+N_3+N_4+N_5=$

1. 210
2. 252
3. 126
4. 125

Q. There are five letters and five addressed envelopes corresponding to them. The number of ways the letters can be placed (one in each envelope) such that no letter is in its corresponding envelope is

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. 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. 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. Let $N=2016$, then the sum of even divisors of $N$ which are not divisible by $16$ is

Q. Let $P=780$, then the correct option(s) among the following is/are

1. Sum of even divisors of $P$ is $1008$
2. Sum of odd divisors of $P$ is $336$
3. Total number of ways in which $P$ can be represented as product of $2$ coprime numbers is $8$
4. Total number of ways in which $P$ can be represented as product of $2$ natural numbers is $12$

Q. A box contains two distinct white balls, three distinct black balls and four distinct red balls, In how many ways can three balls be drawn from the box if at least one black ball is to be included in the drawn

Q. If $P=2^4\cdot 6^3\cdot 5^2\cdot {15}^2$, then the number of proper odd divisors of $P$ will be

1. $29$
2. $30$
3. $59$
4. $60$

Q. A set contains $(2n+1)$ elements. Then the number of subsets of the set which contains at most $n$ elements is

1. $2^n$
2. $2^{2n}$
3. $2^{4n}$
4. $2^{3n}$

Q. Let $3600=x\cdot y$, where $x$ and $y$ are natural numbers, then the possible pairs $(x,y)$ is

1. $21$
2. $23$
3. $45$
4. $46$

Q.

The total number of ways of dividing 15 different things into groups of 8, 4 and 3 respectively is

1. $=\frac{15!}{8!4!3!}$

2. $=\frac{15!}{8!4!}$

3. $=\frac{15!}{8!3!}$

4. $=\frac{15!}{4!3!}$

Q. If $N$ is the number of ways in which a person can walk up a stairway which has $7$ steps if he can take $1$ or $2$ steps up the stairs at a time, then the value of $N/3$ is

Q. A man has $3$ friends. If $N$ is 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, then the value of $N/170$ is

Q. The no of ways of selecting atleast one letter from the letters of the word "PROPORTION" is

1. $288$
2. $286$
3. $289$
4. $287$

Q. The number of ways of selecting atleast one letter from the letters of the word "PROPORTION" is

1. $286$
2. $287$
3. $288$
4. $289$