Permutations and Combinations

The number of ways of getting a sum 16 on throwing a dice four times is ________.

60 words can be made using all the letters of the word \(\mathrm{BHBJO}\), with or without meaning. If these words are written as in a dictionary, then the \(50^{\text {th }}\) word is:

Let the set \(S=\{2,4,8,16, \ldots, 512\}\) be partitioned into 3 sets \(A, B, C\) with equal number of elements such that \(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\mathrm{S}\) and $$\mathrm{A} \cap \mathrm{B}=\mathrm{B} \cap \mathrm{C}...

There are 5 points \(P_1, P_2, P_3, P_4, P_5\) on the side \(A B\), excluding \(A\) and \(B\), of a triangle \(A B C\). Similarly there are 6 points \(\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}\) on the side \(\mathrm{BC}\) and 7 p...

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is ________.

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to __________.

If for some \(m, n ;{ }^6 C_m+2\left({ }^6 C_{m+1}\right)+{ }^6 C_{m+2}>{ }^8 C_3\) and \({ }^{n-1} P_3:{ }^n P_4=1: 8\), then \({ }^n P_{m+1}+{ }^{\mathrm{n}+1} C_m\) is equal to

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : \(A, B\) and \(C\). A student is required to attempt total 15 questions taking at least 4 questions from e...

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is _________.

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

Let \(\alpha=\frac{(4 !) !}{(4 !)^{3 !}}\) and \(\beta=\frac{(5 !) !}{(5 !)^{4 !}}\). Then :

The number of elements in the set $\mathrm{S}=\{(x, y, z): x, y, z \in \mathbf{Z}, x+2 y+3 z=42, x, y, z \geqslant 0\}$ equals __________.

If $\mathrm{n}$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $\mathrm{n}$ is equal to :

Let the number of elements in sets \(A\) and \(B\) be five and two respectively. Then the number of subsets of \(A \times B\) each having at least 3 and at most 6 elements is :

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is :

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is :

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which \(\mathrm{C}\) and \(\mathrm{S}\) do not come together, is \((6 !) \mathrm{k}\), then \(\mathrm{k}\) is equal to :

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ___________.

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is __________.

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is :

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to ____________.

Let 5 digit numbers be constructed using the digits \(0,2,3,4,7,9\) with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is __________.

If ${ }^{2 n+1} \mathrm{P}_{n-1}:{ }^{2 n-1} \mathrm{P}_{n}=11: 21$,
then $n^{2}+n+15$ is equal to :

Let $\mathrm{A}=\left[\mathrm{a}_{i j}\right], \mathrm{a}_{i j} \in \mathbb{Z} \cap[0,4], 1 \leq i, j \leq 2$.
The number of matrices A such that the sum of all entries is a prime number $\mathrm{p} \in(2,13)$ is __________.

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ___________.

The number of seven digits odd numbers, that can
be formed using all theseven digits 1, 2, 2, 2, 3, 3,
5 is ____________.

The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots ., 100\}$ and $b \in\{1,3,5, \ldots . ., 99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is :

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____________.

If all the six digit numbers \(x_1\,x_2\,x_3\,x_4\,x_5\,x_6\) with \(0< x_1 < x_2 < x_3 < x_4 < x_5 < x_6\) are arranged in the increasing order, then the sum of the digits in the \(\mathrm{72^{th}}\) number is _____________.

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is __________.

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is :

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

Let \(x\) and \(y\) be distinct integers where \(1 \le x \le 25\) and \(1 \le y \le 25\). Then, the number of ways of choosing \(x\) and \(y\), such that \(x+y\) is divisible by 5, is ____________.

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then...

A triangle is formed by X-axis, Y-axis and the line \(3x+4y=60\). Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is ____________.

\(\sum\limits_{k = 0}^6 {{}^{51 - k}{C_3}}\) is equal to :

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :

The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ______________.

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is __________

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is :

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is :

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is ___________.

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ____________.

The value of \(\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}\) is :

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ____________.

Number of integral solutions to the equation \(x+y+z=21\), where \(x \ge 1,y\ge3,z\ge4\), is equal to ____________.

A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of tr...

The total number of three-digit numbers, divisible by 3, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is :

The number of seven digit positive integers formed using the digits \(1,2,3\) and \(4\) only and sum of the digits equal to \(12\) is ___________.