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Consider the set of all triangles $O P Q$ where $O$ is origin and $P, Q$ are distinct points in the $x y$ -plane with non-negative integer coordinates $(x, y)$ [i.e. both $x$ and $y$ coordinates are integers] such that $5 x + y = 99$ . Number of such distinct triangles whose area is positive integer, is equal to._____

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Important Questions on Permutation and Combination

EASY

Mathematics>Algebra>Permutation and Combination>Combinations

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 :

EASY

Mathematics>Algebra>Permutation and Combination>Combinations

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

and

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

,

then

n =

EASY

Mathematics>Algebra>Permutation and Combination>Combinations

A password is set with

3

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

EASY

Mathematics>Algebra>Permutation and Combination>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 :

HARD

Mathematics>Algebra>Permutation and Combination>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:

MEDIUM

Mathematics>Algebra>Permutation and Combination>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>Permutation and Combination>Combinations

The number of diagonals of a polygon with

15

sides is

MEDIUM

Mathematics>Algebra>Permutation and Combination>Combinations

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

, then

n

satisfies the equation:

HARD

Mathematics>Algebra>Permutation and Combination>Combinations

If in a regular polygon the number of diagonals is

54

, then the number of sides of this polygon is:

HARD

Mathematics>Algebra>Permutation and Combination>Combinations

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

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

HARD

Mathematics>Algebra>Permutation and Combination>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>Permutation and Combination>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 :

MEDIUM

Mathematics>Algebra>Permutation and Combination>Combinations

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>Combinations

In a group of

6

boys and

4

girls, a team consisting of four children is formed such that the team has atleast one boy. The number of ways of forming a team like this is

EASY

Mathematics>Algebra>Permutation and Combination>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

HARD

Mathematics>Algebra>Permutation and Combination>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} =

HARD

Mathematics>Algebra>Permutation and Combination>Combinations

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>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:

EASY

Mathematics>Algebra>Permutation and Combination>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

MEDIUM

Mathematics>Algebra>Permutation and Combination>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

Consider the set of all triangles OPQ where O is origin and P,Q are distinct points in the xy-plane with non-negative integer coordinates x, y [i.e. both x and y coordinates are integers] such that 5x+y=99. Number of such distinct triangles whose area is positive integer, is equal to._____

Important Questions on Permutation and Combination

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