Number of ways of selecting three integers from 1 , 2 , 3 , … , n if their sum is divisible by 3 is

2 n - 1 3 C 3 + n + 2 3 C 3 + n - 1 3 2 n + 2 3 if n = 3 k + 2 , k ∈ N .

A is a set containing n elements. A subset P 1 of A is chosen. The set A is reconstructed by replacing the elements of P 1 . Next, a subset P 2 of A is chosen and again the set is reconstructed by replacing the elements of P 2 . In this way, m > 1 subsets P 1 , P 2 , … . , P m of A are chosen. The number of ways of choosing P 1 , P 2 , … , P m is

2 m - 1 n if P 1 ∪ P 2 ∪ … ∪ P m = A .

2 m n if P 1 ∪ P 2 ∪ … ∪ P m = A .

2 m n if P 1 ∩ P 2 ∩ … ∩ P m = ϕ .

If 10 ! = 2 p · 3 q · 5 r · 7 s , then

number of ways of putting 10 ! as a product of two natural numbers is 135 .

number of divisors of 10 ! is 280 .

HARD

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IMPORTANT

Earn 100

Let $f (n)$ be the number of region in which $n$ coplanar circles can divide the plane. If it is known that each pair of circles intersect in two different points and no three of them have common point of intersection, then

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Important Questions on Permutation and Combination

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 10

Given that the divisors of

n = 3^{p} \cdot 5^{q} \cdot 7^{r}

are of the form

4 λ + 1, λ \geq 0

. Then

HARD

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 11

Number of ways of selecting three integers from

\{1, 2, 3, \dots, n\}

if their sum is divisible by

3

HARD

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 12

Number of points of intersection of

n

straight lines if

n

satisfies

{}^{n + 5}P_{n + 1} = \frac{11 (n - 1)}{2} \times^{n + 3} P_{n}

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 13

Number of the shortest ways in which we can reach from the point

(0, 0, 0)

to point

(3, 7, 11)

in space where the movement is possible only among the

x

-axis,

y

-axis and

z

-axis or parallel to them and change of axes is permitted only at integral points (an integral point is one which has its coordinates as integer) is

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 14

Number of ways in which

30

identical things are distributed among six persons is

MEDIUM

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 15

N

denotes the number of ways of selecting

r

objects out of

n

distinct objects

(r \geq n)

with unlimited repetition but with each object included at least once in selection, then

N

is equal to

HARD

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 16

A

is a set containing

n

elements.

A

subset

P_{1}

A

is chosen. The set

A

is reconstructed by replacing the elements of

P_{1}

. Next, a subset

P_{2}

A

is chosen and again the set is reconstructed by replacing the elements of

P_{2}

. In this way,

m (> 1)

subsets

P_{1}, P_{2}, \dots ., P_{m}

A

are chosen. The number of ways of choosing

P_{1}, P_{2}, \dots, P_{m}

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 17

10! = 2^{p} \cdot 3^{q} \cdot 5^{r} \cdot 7^{s}

, then

Let fn be the number of region in which n coplanar circles can divide the plane. If it is known that each pair of circles intersect in two different points and no three of them have common point of intersection, then

Important Questions on Permutation and Combination

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Let $f (n)$ be the number of region in which $n$ coplanar circles can divide the plane. If it is known that each pair of circles intersect in two different points and no three of them have common point of intersection, then