Permutations & Combinations Questions with Solutions

There are 6 tasks and 6 persons. Task 1 cannot be assigned to person 1 or person 2. Task 2 must be assigned to either person 3 or person 4. Every person is to be given exactly one task. In how many ways can the assignment be done?

How many different paths in the $xy$ -plane are there from $\left(1,3\right)$ to $\left(5,6\right)$, if a path proceeds one step at a time either right (R) or upward (U)?

There are 10 points in a plane, out of which 6 are collinear (no other three are collinear). The number of triangles formed by joining these points is:

Twelve villages in a district are divided into 3 zones with 4 villages per zone. The telephone department intends to connect the villages so that every two villages in the same zone are connected with three direct lines and every two villages belonging to different zones are connected with two direct lines. How many direct lines are required?

Using the digits 1, 5, 2, 8 (each exactly once), all possible four-digit numbers are formed. The sum of all such numbers lies between:

All the letters of the word INDIA are arranged in all possible ways and the words so formed are written in dictionary order. What is the 58th word?

The number of functions $f$ from $A=0,1,2$ into $B=0,1,2,3,4,5,6,7$ such that $f\left(i\right)\le f\left(j\right)$ for $i<j$ is

An eight digit number divisible by $9$ is to be formed by using $8$ digits out of the digits $0,1,2,\dots ,9$ without replacement. The number of ways in which this can be done is

In an objective type examination, $120$ objective type questions are there, each with $4$ options P, Q, R and S. A candidate can choose either one of these options or can leave the question unanswered. How many different ways exist for answering this question paper?

$12$ members were present at a board meeting. Each member shook hands with all of the other members before and after the meeting. How many handshakes were there?

How many words can be formed starting with letter D taking all letters from the word DELHI so that the letters are not repeated:

Naresh has 10 friends, and he wants to invite 6 of them to a party. How many times will 3 particular friends never attend the party?

There is a young boy’s birthday party in which 3 friends have attended. The mother has arranged 10 games where a prize is awarded for a winning game. The prizes are identical. If each of the 4 children receives at least one prize, then how many distributions of prizes are possible?

If $\frac{n!}{2!\left(n-2\right)!}$ and $\frac{n!}{4!\left(n-4\right)!}$ are in the ratio $2:1$, then the value of $n$ is

If $n$ is an integer between 0 and 21, then the value of $n$ for which $n!\left(21-n\right)!$ is minimum is

There are 50 questions in a paper. The number of ways in which a student can attempt one or more questions is

A password consists of two alphabets from English followed by three numbers chosen from 0 to 3. If repetitions are allowed, the number of different passwords is

The number of ways in which 5 days can be chosen in each of the 12 months of a non-leap year, is:

The number of ways to arrange the letters of the English alphabet, so that there are exactly 5 letters between a and b, is:

If n and r are integers such that 1 ≤ r ≤ n, then the value of $n\cdot {}^{n-1}{C}_{r-1}$ is