Permutations and Combinations

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits \(1,2,3,4,5\) with repetition, is _________.

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

Let the digits a, b, c be in A. P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits \(0,1,3,5,7\) and 9 without repetition, is equal to :

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is _________.

The number of triplets \((x, \mathrm{y}, \mathrm{z})\), where \(x, \mathrm{y}, \mathrm{z}\) are distinct non negative integers satisfying \(x+y+z=15\), is :

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is :

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons, who participated in the tournament, is ___________.

The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ___________.

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to __________.

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is :

The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________.

Let b1b2b3b4 be a 4-element permutation with bi \(\in\) {1, 2, 3, ........, 100} for 1 \(\le\) i \(\le\) 4 and bi \(\ne\) bj for i \(\ne\) j, such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then ...

The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to ____________.

The number of matrices of order \(3 \times 3\), whose entries are either 0 or 1 and the sum of all the entries is a prime number, is __________.

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits \(2,3,4,5,6\) (repetition of digits is not allowed) and divisible by 55 is _________.

The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is :

The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives at least 4 and at most 7 candies, C3 receives at least 2 and at most 6 candies, is equal to :

Let \(S\) be the set of all passwords which are six to eight characters long, where each character is either an alphabet from \(\{A, B, C, D, E\}\) or a number from \(\{1,2,3,4,5\}\) with the repetition of characters allowed. If the number ...

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168 , then \(\mathrm{b}+3 \mathrm{~g}\) is equal to ____________.

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _____________.

Let A be a matrix of order 2 \(\times\) 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is ___________.

There are ten boys B1, B2, ......., B10 and five girls G1, G2, ........, G5 in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B1 and B2 together should not be the members of a group, is...

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ___________.

The number of 5-digit natural numbers, such that the product of their digits is 36 , is __________.

Numbers are to be formed between 1000 and 3000 , which are divisible by 4 , using the digits \(1,2,3,4,5\) and 6 without repetition of digits. Then the total number of such numbers is ____________.

Let A be a 3 \(\times\) 3 matrix having entries from the set {\(-\)1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ___________.

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.

The total number of three-digit numbers, with one digit repeated exactly two times, is ______________.

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____________.

In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, \(-\)2 marks for each wrong answer and 0 mark if the question is not attempted. Then, t...

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____________.

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________.

Let n be a non-negative integer. Then the number of divisors of the form "4n + 1" of the number (10)10 . (11)11 . (13)13 is equal to __________.

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ____________.

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A \(\subseteq\) S : A \(\ne\) \(\phi\) and the sum of all the elements of A is not a multiple of 3} is _______________.

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :

A natural number has prime factorization given by n = 2x3y5z, where y and z are such that y + z = 5 and y\(-\)1 + z\(-\)1 = \({5 \over 6}\), y > z. Then the number of odd divisions of n, including 1, is :

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ______________.

If \({}^1{P_1} + 2.{}^2{P_2} + 3.{}^3{P_3} + .... + 15.{}^{15}{P_{15}} = {}^q{P_r} - s,0 \le s \le 1\), then \({}^{q + s}{C_{r - s}}\) is equal to ______________.

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the ...

If \({}^n{P_r} = {}^n{P_{r + 1}}\) and \({}^n{C_r} = {}^n{C_{r - 1}}\), then the value of r is equal to :

The total number of positive integral solutions (x, y, z) such that xyz = 24 is :

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is _____________.

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at
least 2 Indians and double the number of foreigners as Indians. Then the number of ways,
the committee can be formed, is :

The students S1, S2, ....., S10 are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is ___________.

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to _____________.

There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsman and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsman and 1 wicketkeeper, is ...

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 n...

Let P1, P2, ......, P15 be 15 points on a circle. The number of distinct triangles formed by points Pi, Pj, Pk such that i +j + k \(\ne\) 15, is :