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

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Important Questions on Permutations and Combinations

EASY

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

If the number of five digit numbers with distinct digits and

2

at the

10^{t h}

place is

336 k

, then

k

is equal to:

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Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

The number of natural numbers less than

7000

which can be formed by using the digits

0, 1, 3, 7, 9

(repetition of digits allowed) is equal to:

EASY

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

A committee of five members is to be formed out of

3

trainees,

4

professors and

6

research associates. In how many different ways can this be done if the committee should have all the

4

professors and

1

research associate or all

3

trainees and

2

professors?

EASY

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Let

M = \{(a_{1}, a_{2}, a_{3}) : a_{i} \in \{1, 2, 3, 4\}, a_{1} + a_{2} + a_{3} = 6\} .

Then the number of elements in

M

EASY

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

S

is a set with

10

elements and

A = \{(x, y) : x, y \in S, x \neq y\}

,

then the number of elements in

A

EASY

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Let

S = {0, 1, 2, 3, \dots, 100}

. The number of ways of selecting

x, y \in S

such that

x \neq y

and

x + y = 100

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

If the four letter words (need not be meaningful) are to be formed using the letters from the word "MEDITERRANEAN" such that the first letter is

R

and the fourth letter is

E

, then the total number of all such words is :

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

The chairs at an auditorium are to be labelled with a letter and a positive integer not exceeding

100

. The largest number of chairs that can be marked differently is equal to

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

The number of integers

n

with

100 \leq n \leq 999

and containing at most two distinct digits is

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Let

S = \{(a, b) : a, b \in Z, 0 \leq a, b \leq 18\} .

The number of elements

(x, y)

S

such that

3 x + 4 y + 5

is divisible by

19

is,

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Let

m

(respectively,

n

) be the number of

5

-digit integers obtained by using the digits

1, 2, 3, 4, 5

with repetitions (respectively, without repetitions) such that the sum of any two adjacent digits is odd. Then

\frac{m}{n}

is equal to

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

The number of numbers between

2, 000

and

5, 000

that can be formed with the digits

0, 1, 2, 3, 4

(repetition of digits is not allowed) and are multiple of

3

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

A five-digit number divisible by

3

is to be formed using the numbers

0, 1, 2, 3, 4

and

5

without repetition. The total number of ways this can be done is

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Five points are marked on a circle. The number of distinct polygons of three or more sides can be drawn using some (or all) of the five points as vertices is

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

The number of ways of dividing

15

men and

15

women into

15

couples, each consisting of man and woman is

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

The number of times the digit

7

will be written when listing the integers from

1

1000

HARD

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Consider a rectangle

A B C D

having

5, 6, 7, 9

points in the interior of the line segments

A B, B C, C D, D A

respectively. Let

α

be the number of triangles having these points from different sides as vertices and

β

be the number of quadrilaterals having these points from different sides as vertices. Then

(β - α)

is equal to

HARD

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Let

S = \{(a, b)| a, b \in Z, 0 \leq a, b \leq 18\}

. The number of lines in

R^{2}

passing through

(0, 0)

and exactly one other point in

S

is-

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

Ten points lie in a plane so that no three of them are collinear. The number of lines passing through exactly two of these points and dividing the plane into two regions each containing four of the remaining points is

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Fundamental Principle of Counting

The number of integers greater than

6000

that can be formed, using the digits

3, 5, 6, 7

and

8

, without repetition is

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

Important Questions on Permutations and Combinations

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