Permutations and Combinations

Consider a number lock with three rings. The first ring is marked with digits 0 to 9. The second ring is marked with prime numbers greater than 2 but less than 30. The third ring is marked with all vowels. Determine the total number of unsuccessful attempts to open the lock.

Given the digits 3, 4, 5, 6, 7, and 8, determine the number of possible numerical values greater than 750,000 that can be formed if repetition is not allowed.

Assuming no two consecutive digits are the same, derive a closed-form expression for the number of $n$-digit numbers that can be formed.

Given the digits 1, 2, 3, 4, 6, and 7, determine the number of three-digit even numbers that can be formed if no digit is repeated.

Determine the number of words that can be formed from the letters of the word 'MONDAY' in which the first letter is a vowel.

Determine the total number of ways of answering 5 objective-type questions, each having 4 choices, considering that the number of choices includes a 'none of these' choice.

How many different signals can be made by hoisting 5 flags from 8 flags of different colours?

In how many ways can 3 distinct prizes be distributed among 4 distinct boys, if no boy gets more than one prize?

In how many ways can 3 prizes be distributed among 4 boys if each boy is eligible to receive any number of prizes?

In a group of boys, the number of arrangements of 4 boys is 12 times the number of arrangements of 2 boys. What is the total number of boys in the group?

If $^nP_3 = 20 \cdot ^nP_3$, what is the value of 'n'?

At an election, a voter may vote for any number of candidates not greater than the number to be elected. If there are 10 candidates and 4 are to be elected, and a voter votes for at least one candidate, in how many ways can the voter cast their vote?

A committee consisting of 2 men and 2 women is to be chosen from 5 men and 6 women. In how many ways can you do this?

A father with 7 children takes 4 at a time to a zoo, as often as he can, without taking the same four children together more than once. How often will he go and how often will each child go?

In how many ways can the letters of the word TRIANGLE be arranged such that no two vowels occur together?

From seven consonants and four vowels, derive the number of six-letter words can be formed by taking four consonants and two vowels, assuming that each ordered group of letters constitutes a valid word.

In how many ways can 22 different books be distributed among 5 students, such that two students receive 5 books each, and the remaining students get 4 books each?

The number of groups that can be made from 5 different green balls, 4 different blue balls, and 3 different red balls, if at least 1 green and 1 blue ball is included, is?

In how many ways can at least one horse and at least one dog be selected from eight horses and seven dogs?

A shopkeeper has 10 copies of each of nine different books. Determine the number of ways in which at least one book can be selected.

Determine the number of proper divisors of $2^p \cdot 6^q \cdot 15^r$, where p, q, and r are prime integers.

How many words can be made by rearranging the letters of the word "APURBA" such that vowels and consonants alternate?

How many words can be formed from the letters of the word 'INSTITUTION' in which the first two letters are 'N'?

Determine the number of different words that can be formed from the letters of the word APPLICATION such that two vowels never come together.

Six identical coins are arranged in a row. Find the number of ways in which the number of tails is equal to the number of heads.

Determine the number of permutations that can be formed by arranging all the letters of the word 'NINETEEN' in which no two E's occur together.

If there are (n+1) white balls and (n+1) black balls, each set numbered from 1 to n+1, in how many ways can the balls be arranged such that adjacent balls are of different colours?

If the letters of the word 'SACHIN' are arranged in all possible ways and these words are written out as in a dictionary, what is the serial number at which the word 'SACHIN' appears?

If eleven members of a committee sit at a round table, with the President and Secretary always sitting together, what is the number of possible seating arrangements?

In how many ways can a garland be made from exactly 10 flowers?

There are 5 gentlemen and 4 ladies prepared to dine at a round table. How many ways can they seat themselves so that no two ladies are seated together?

In how many distinct ways can 4 boys and 4 girls stand in a circle so that each boy and each girl is one after the other?

Determine the number of straight lines joining any two of twelve points in a plane if three of the points are collinear.

If 'm' parallel lines in a plane are intersected by a family of 'n' parallel lines, how many parallelograms are formed?

Determine all positive integer solutions to the equation $x + y + z = 12$.

If $x + y + z + w 20$, find the number of non-negative integral solutions.

In how many ways can 25 identical balls be distributed among Ram, Shyam, Sunder, and Ghanshyam such that at least 1, 2, 3, and 4 balls are given to Ram, Shyam, Sunder, and Ghanshyam, respectively?

A person writes letters to 6 friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least 4 of them are in the wrong envelopes?

If there are four balls of different colours and four boxes of the same colours as the balls, in how many ways can the balls be placed in the boxes, one ball in each box, such that no ball goes into the box of the same colour?

Calculate number of shortest paths possible between A and B vertices in the following 2-D grid (assuming that you can only travel upwards or to the right)

In how many ways can a person go from A to B if they can only travel upwards and to the right?