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If ${}^{43}C_{r - 6} =^{43} C_{3 r + 1}$ , then the value of $r$ is

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Important Questions on Counting Principles

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Aptitude>Arithmetic>Counting Principles>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

Aptitude>Arithmetic>Counting Principles>Combinations

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

then

k

equals

HARD

Aptitude>Arithmetic>Counting Principles>Combinations

If in a regular polygon the number of diagonals is

54

, then the number of sides of this polygon is:

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Aptitude>Arithmetic>Counting Principles>Combinations

The value of

\sum_{r = 1}^{10} r \cdot \frac{{}^{n}C_{r}}{{}^{n}C_{r - 1}}

is equal to

EASY

Aptitude>Arithmetic>Counting Principles>Combinations

The value of

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

EASY

Aptitude>Arithmetic>Counting Principles>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:

EASY

Aptitude>Arithmetic>Counting Principles>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 :

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Aptitude>Arithmetic>Counting Principles>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

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Aptitude>Arithmetic>Counting Principles>Combinations

The number of selection of

n

objects from

2 n

objects of which

n

are identical and the rest are different, is

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Aptitude>Arithmetic>Counting Principles>Combinations

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

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Aptitude>Arithmetic>Counting Principles>Combinations

Consider a class of

5

girls and

7

boys. The number of different teams consisting of

2

girls and

3

boys that can be formed from this class, if there are two specific boys

A

and

B,

who refuse to be the members of the same team, is:

EASY

Aptitude>Arithmetic>Counting Principles>Combinations

Total number of

6

-digit numbers in which only and all the five digits

1, 3, 5, 7

and

9

appears, is

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Aptitude>Arithmetic>Counting Principles>Combinations

S = C_{1} + 2 \cdot C_{2} + 3 \cdot C_{3} + \dots + n \cdot C_{n}

, then

S

is equal to

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Aptitude>Arithmetic>Counting Principles>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:

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Aptitude>Arithmetic>Counting Principles>Combinations

The number of integers greater than

6000

that can be formed, using the digits

3, 5, 6, 7

and

8

, without repetition is

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Aptitude>Arithmetic>Counting Principles>Combinations

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

Aptitude>Arithmetic>Counting Principles>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 :

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Aptitude>Arithmetic>Counting Principles>Combinations

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

Aptitude>Arithmetic>Counting Principles>Combinations

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

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Aptitude>Arithmetic>Counting Principles>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

If Cr-643=43C3r+1, then the value of r is

Important Questions on Counting Principles

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If ${}^{43}C_{r - 6} =^{43} C_{3 r + 1}$ , then the value of $r$ is