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 .

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 .

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Number of ways of selecting three integers from $\{1, 2, 3, \dots, n\}$ if their sum is divisible by $3$ is

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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 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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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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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

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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

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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}

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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

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 18

P = 21 (21^{2} - 1^{2}) (21^{2} - 2^{2}) (21^{2} - 3^{2}) \dots (21^{2} - 10^{2})

, then

P

is divisible by

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 7 - Permutation and Combination>Exercises>Q 19

Let

n

be a four-digit integer in which all the digits are different. If

x

is a number of odd integers and

y

is a number of even integers, then

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

Important Questions on Permutation and Combination

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Number of ways of selecting three integers from $\{1, 2, 3, \dots, n\}$ if their sum is divisible by $3$ is