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Let $S$ represents the number of all $6$ digit numbers such that
(i) All the digits of each number are from the set ${1, 2, 3, 4, 5}$ .
(ii) Any digit that appears in the number appears at least twice. Find the sum of digits of $S$ .

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Important Questions on Permutations and Combinations

EASY

Mathematics>Algebra>Permutations and Combinations>Combinations

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 :

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Combinations

The number of diagonals of a polygon with

15

sides is

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Combinations

The least value of a natural number

n

such that

(\frac{n - 1}{5}) + (\frac{n - 1}{6}) < (\frac{n}{7})

, where

(\frac{n}{r}) = \frac{n!}{(n - r)! r!}

, is

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Combinations

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>Permutations and Combinations>Combinations

If in a regular polygon the number of diagonals is

54

, then the number of sides of this polygon is:

EASY

Mathematics>Algebra>Permutations and Combinations>Combinations

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

EASY

Mathematics>Algebra>Permutations and Combinations>Combinations

A password is set with

3

distinct letters from the word LOGARITHMS. How many such passwords can be formed?

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Combinations

A committee of

11

member is to be formed from

8

males and

5

females. If

m

is the number of ways the committee is formed with at least

6

males and

n

is the number of ways the committee is formed with at least

3

females, then

HARD

Mathematics>Algebra>Permutations and Combinations>Combinations

The number of noncongruent integer-sided triangles whose sides belong to the set

{10, 11, 12, . . . . . ., 22}

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Combinations

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:

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Combinations

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

, then

n

satisfies the equation:

EASY

Mathematics>Algebra>Permutations and Combinations>Combinations

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

HARD

Mathematics>Algebra>Permutations and Combinations>Combinations

The value of

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

is equal to:

HARD

Mathematics>Algebra>Permutations and Combinations>Combinations

In a tournament with five teams, each team plays against every other team exactly once. Each game is won by one of the playing teams and the winning team scores one point, while the losing team scores zero. Which of the following is NOT necessarily true?

HARD

Mathematics>Algebra>Permutations and Combinations>Combinations

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} =

EASY

Mathematics>Algebra>Permutations and Combinations>Combinations

The Number of ways of choosing

10

objects out of

31

objects of which

10

are identical and the remaining

21

are distinct, is:

MEDIUM

Mathematics>Algebra>Permutations and Combinations>Combinations

Suppose that

20

pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

EASY

Mathematics>Algebra>Permutations and Combinations>Combinations

{{}^{n}C}_{r - 1} = 36, {{}^{n}C}_{r} = 84

and

{{}^{n}C}_{r + 1} = 126

,

then

n =

HARD

Mathematics>Algebra>Permutations and Combinations>Combinations

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:

HARD

Mathematics>Algebra>Permutations and Combinations>Combinations

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 :

Let S represents the number of all 6 digit numbers such that (i) All the digits of each number are from the set {1,2,3,4,5}. (ii) Any digit that appears in the number appears at least twice. Find the sum of digits of S.

Important Questions on Permutations and Combinations

Learn and Prepare for any exam you want !

Let $S$ represents the number of all $6$ digit numbers such that
(i) All the digits of each number are from the set ${1, 2, 3, 4, 5}$ .
(ii) Any digit that appears in the number appears at least twice. Find the sum of digits of $S$ .