Fundamental Principle of Counting

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JEE Main 2015

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The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is :

Let $X = {1, 2, 3, 4, 5}$ . The number of different ordered pairs (Y,Z) that can formed such that $Y \subseteq X, Z \subseteq X$ and $Y \cap Z$ is empty is:

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are

Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is

If the LCM of $p, q$ is $r^{2} s^{4} t^{2}$ , where $r, s, t$ are prime numbers and $p, q$ are the positive integers then the number of ordered pair $(p, q)$ is

A rectangle with sides of length $(2 m - 1)$ and $(2 n - 1)$ units is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is

(A)

(m + n - 1)^{2}

(B)

4^{m + n - 1}

(C)

m^{2} n^{2}

(D)

m (m + 1) n (n + 1)

JEE Advanced 1998

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Number of divisor of the form $4 n + 2$ ( $n \geq 0$ ) of the integer 240 is

An $n$ -digit number is a positive number with exactly $n$ digits. Nine hundred distinct $n$ -digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of $n$ for which this is possible is

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are

In a certain test, $a_{i}$ students gave wrong answers to atleast $i$ questions, where $i = 1, 2, \dots, k$ . No student gave more than $k$ wrong answers. The total number of wrong answers given is .........

JEE Advanced 2012

LEVELJEE Main

Comprehension Passage

Let

a_{n}

denote the number of all

n

-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let

b_{n}

= the number of such

n

-digit integers ending with digit 1 and

c_{n}

= the number of such

n

-digit integers ending with digit 0.

Question 1:

The value of $b_{6}$ is

Which of the following is correct?

(A)

a_{17} = a_{16} + a_{15}

(B)

c_{17} \neq = c_{16} + c_{15}

(C)

b_{17} \neq = b_{16} + c_{16}

(D)

a_{17} = c_{17} + b_{16}

JEE Main 2018 (15 April Shift 1)

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n-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is

JEE Main 2019 (9 January)

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The number of natural numbers less than 7,000 which can be formed by using the digits 0,1,3,7,9 (repetition of digits allowed) is equal to :

JEE Main 2019 (08 April Shift 2)

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The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :

JEE Main 2020 - 9 Jan (Morning)

LEVELJEE Main

If number of 5 digit numbers which can be formed without repeating any digit while tenth place of all of the numbers must be 2 is 336 k find value of k

JEE Main 2021 (27 Aug Shift 1)

LEVELJEE Main

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 .

JEE Main 2021 (22 July Shift 1)

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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

JEE Main 2021 (27 July Shift 2)

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Let $n$ be a non-negative integer. Then the number of divisors of the form " $4 n + 1$ " of the number $(10)^{10} \cdot (11)^{11} \cdot (13)^{13}$ is equal to

JEE Main 2021 (25 February Shift 1)

LEVELJEE Main

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

JEE Main 2021 (March) (17 March Shift 1)

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Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

JEE Main 2021 (March) (18 March Shift 1)

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The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

JEE Main 2022 (25 June Shift 1)

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

JEE Main 2022 (29 June Shift 1)

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Let $b_{1}, b_{2}, b_{3}, b_{4}$ be a 4-element permutation with $b_{i} \in {1, 2, 3, \dots, 100}$ for $1 \leq i \leq 4$ and $b_{i} \neq = b_{j}$ for $i \neq = j$ , such that either $b_{1}, b_{2}, b_{3}$ are consecutive integers or $b_{2}, b_{3}, b_{4}$ are consecutive integers. Then the number of such permutations $b_{1}, b_{2}, b_{3}, b_{4}$ is equal to ______.

JEE Main 2022 (29 June Shift 2)

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

JEE Main 2022 (26 July Shift 2)

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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 ______.

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let X={1,2,3,4,5}. The number of different ordered pairs (Y,Z) that can formed such that Y⊆X,Z⊆X and Y∩Z is empty is:

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are

Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the LCM of p,q is r2s4t2, where r,s,t are prime numbers and p,q are the positive integers then the number of ordered pair (p,q) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Number of divisor of the form 4n+2 (n≥0) of the integer 240 is

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of b6​ is

Which of the following is correct?

n-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is

The number of natural numbers less than 7,000 which can be formed by using the digits 0,1,3,7,9 (repetition of digits allowed) is equal to :

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :

If number of 5 digit numbers which can be formed without repeating any digit while tenth place of all of the numbers must be 2 is 336 k find value of k

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 .

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

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

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

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

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

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 $X = {1, 2, 3, 4, 5}$ . The number of different ordered pairs (Y,Z) that can formed such that $Y \subseteq X, Z \subseteq X$ and $Y \cap Z$ is empty is:

If the LCM of $p, q$ is $r^{2} s^{4} t^{2}$ , where $r, s, t$ are prime numbers and $p, q$ are the positive integers then the number of ordered pair $(p, q)$ is

Number of divisor of the form $4 n + 2$ ( $n \geq 0$ ) of the integer 240 is

The value of $b_{6}$ is

Let $n$ be a non-negative integer. Then the number of divisors of the form " $4 n + 1$ " of the number $(10)^{10} \cdot (11)^{11} \cdot (13)^{13}$ is equal to