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Let $a, b, c, d$ be non zero distinct digits. The number of $4$ digit numbers $a b c d$ such that $a b + c d$ is even, is divisible by :

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Important Questions on Permutation and Combination

MEDIUM

Mathematics>Algebra>Permutation and Combination>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:

MEDIUM

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangement is

MEDIUM

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

From

6

different novels and

3

different dictionaries,

4

novels and

1

dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:

HARD

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

{\sum_{i = 1}^{20} (\frac{^{20} C_{i - 1}}{^{20} C_{i} +^{20} C_{i - 1}})}^{3} = \frac{k}{21},

then

k

equals

HARD

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

Let

A = \{x_{1}, x_{2}, \dots, x_{7}\} and B = \{y_{1}, y_{2}, y_{3}\}

be two sets containing seven and three distinct elements respectively. Then the total number of functions

f : A \to B

that are onto, if there exist exactly three elements

x

A

such that

f (x) = y_{2},

is equal to:

MEDIUM

Mathematics>Algebra>Permutation and Combination>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

EASY

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by

66

, then the number of men who participated in the tournament lies in the interval

MEDIUM

Mathematics>Algebra>Permutation and Combination>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 :

HARD

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

If in a regular polygon the number of diagonals is

54

, then the number of sides of this polygon is:

MEDIUM

Mathematics>Algebra>Permutation and Combination>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>Permutation and Combination>Fundamental Principle of Counting

A debate club consists of

6

girls and

4

boys. A team of

4

members is to be selected from this club including the selection of a captain (from among these

4

members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

HARD

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

Let

T_{n}

be the number of all possible triangles formed by joining vertices of an

n

-sided regular polygon. If

T_{n + 1} - T_{n} = 10

, then the value of

n

is :

MEDIUM

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

\frac{^{n + 2} C_{6}}{^{n - 2} P_{2}} = 11

, then

n

satisfies the equation:

EASY

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

Consider three boxes, each containing

10

balls labelled

1, 2, \dots ., 10

. Suppose one ball is randomly drawn from each of the boxes. Denote by

n_{i}

, the label of the ball drawn from the

i^{t h}

box,

(i = 1, 2, 3)

. Then, the number of ways in which the balls can be chosen such that

n_{1} < n_{2} < n_{3}

is :

HARD

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

The value of

\sum_{r = 1}^{15} r^{2} (\frac{^{15} C_{r}}{^{15} C_{r - 1}})

is equal to:

MEDIUM

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

A group of students comprises of

5

boys and

n

girls. If the number of ways, in which a team of

3

students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is

1750,

then

n

is equal to

HARD

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

Let

S = \{1,2, 3, \dots . 9\}

. For

k = 1,2, \dots 5,

let

N_{k}

be the number of subsets of

S

, each containing five elements out of which exactly

k

are odd. Then

N_{1} + N_{2} + N_{3} + N_{4} + N_{5} =

MEDIUM

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

A man

X

has

7

friends,

4

of them are ladies and

3

are men. His wife

Y

also has

7

friends,

3

of them are ladies and

4

are men. Assume

X

and

Y

have no common friends. Then the total number of ways in which

X

and

Y

together can throw a party inviting

3

ladies and

3

men, so that

3

friends of each of

X

and

Y

are in this party is:

EASY

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

The number of ways of selecting

15

teams from

15

men and

15

women, such that each team consists of a man and a woman is

EASY

Mathematics>Algebra>Permutation and Combination>Fundamental Principle of Counting

There are

m

men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by

84

, then the value of

m

is :

Let a, b, c, d be non zero distinct digits. The number of 4 digit numbers abcd such that ab+cd is even, is divisible by :

Important Questions on Permutation and Combination

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Let $a, b, c, d$ be non zero distinct digits. The number of $4$ digit numbers $a b c d$ such that $a b + c d$ is even, is divisible by :