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From a group of $7$ men, and $6$ women, five persons are to be selected to form a committee so that at least $3$ men are there in the committee. In how many ways can it be done?

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Important Questions on Permutation and Combination

MEDIUM

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

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 :

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

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,

EASY

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

The value of

{}^{16}C_{9} + {}^{16}C_{10} - {}^{16}C_{6} - {}^{16}C_{7}

MEDIUM

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

The number of integers

n

with

100 \leq n \leq 999

and containing at most two distinct digits is

EASY

Mathematics>Algebra>Permutation and Combination>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

MEDIUM

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

The number of selection of

n

objects from

2 n

objects of which

n

are identical and the rest are different, is

MEDIUM

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

In order to get through in an examination of nine papers, a candidate has to pass in more papers than the number of papers in which he fails. The number of ways in which he can fail, in this examination is

HARD

Mathematics>Algebra>Permutation and Combination>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-

EASY

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

Total number of

6

-digit numbers in which only and all the five digits

1, 3, 5, 7

and

9

appears, is

EASY

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

MEDIUM

Mathematics>Algebra>Permutation and Combination>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

EASY

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

Let

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

where

C_{r} = {}^{n}C_{r}

and

(C_{0} + C_{1}) (C_{1} + C_{2}) \dots (C_{n - 1} + C_{n}) = A C_{1} C_{2} \dots C_{n}

, then for

n = 5, A

is equal to

EASY

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

In how many ways a team of $5$ members can be chosen from $8$ members?

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

From a group of 7 men, and 6 women, five persons are to be selected to form a committee so that at least 3 men are there in the committee. In how many ways can it be done?

Important Questions on Permutation and Combination

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From a group of $7$ men, and $6$ women, five persons are to be selected to form a committee so that at least $3$ men are there in the committee. In how many ways can it be done?