Permutation Test

Result

Permutation Test - 11

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Weekly Quiz Competition

Question 1
1 / -0

The greatest integer, which divides n! + 1, n ∈ N, n ≥ 3 is

A
None of these

B
n

C
2

D
3

Solution

n! = 1.2.3........................n

∴ n! is divisible by any number from 2 to n.

Consequently n! +1. when divided by any number between 2 and n leaves 1 as remainder. Hence

n! + 1 is not divisible by any of the numbers from 2 to 'n' .

The correct answer is: n
Question 2
1 / -0

A closet has 5 pairs of shoes. The number of ways in which 4 shoes can be chosen from it so that there will be no complete pair is

A
200

B
160

C
None of these

D
80

Solution

Number of selection with no complete pair = total number of ways of choosing 4 shoes - number of ways in which there is at lest one complete pair.

= ¹⁰C₄ - [⁵C₁ × ⁸C₂ - ⁵C₂]

= 210 - [140 - 10] = 80

The correct answer is: 80
Question 3
1 / -0

Number of ways of selecting n things out of 3n things of which n are of one kind and alike and n are alike of 2^nd kind and rest are unlike is

A
(n + 2) 2^(n–1)

B
³ⁿC_n – ²ⁿC_n

C
3nC_n

D
3n

Solution

Number of ways coefficient of xⁿ in

(1 + x + x² + x³ + ......+ xⁿ)² (1 + x)ⁿ

= coefficient of xⁿ in (1- xⁿ⁺¹)²(1 - x)^-2(1 - x)

= coefficient of xⁿ in (1 - x)^-2[(2 - (1 - x))]ⁿ

= coefficient of xⁿ in 2ⁿ (1 - x)^-2) - 2ⁿ^-¹.ⁿC₁ (1 - x)^-1 + 2ⁿ^-².ⁿC₂ - 2^n-3.ⁿC₃ (1 - x) + ......

= 2ⁿ(n + 1) - 2ⁿ^-¹.n

= 2ⁿ^-¹(n + 2).

The correct answer is: (n + 2) 2⁽ⁿ^–¹⁾
Question 4
1 / -0

How many different 5 letter sequences can be made using the letters A,B,C,D with repetition such that the sequence does not include the word BAD?

A
48

B
658

C
976

D
1048

Solution

Number of sequences can be formed = 4⁵

Number of sequences including BAD = 3.4²

Total number of sequences not of having BAD = 4⁵ - 3.4² = 976.

The correct answer is: 976
Question 5
1 / -0

The number of ways in which the sequence of a 8×8 chess board can be painted red or blue so that each 2×2 squares has two red and two blue squares is

A
2⁹ – 2

B
2⁹ – 1

C
2⁹ – 4

D
2⁹

Solution

Number of ways when the squares are alternating colour in first column is 2⁸.

Number of ways when the squares in the first column are not alternating colour = 2⁸ – 2

⇒ total number of ways = 2⁸ + 2⁸ – 2 = 2⁹ – 2

The correct answer is: 2⁹ – 2
Question 6
1 / -0

Statement-1 : Number of permutations of "n" dissimilar things taken "n" at a time is ⁿp_n.

Statement-2 : n(A) = n(B) = n then the total number of functions from A to B are n!

A
Both Statements-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1

B
Both Statements-1 and Statement-2 are true and Statement-2 is not the correct explanation of Statement-1

C
Statement-1 is true and Statement-2 is false

D
Statement-1 is false and Statement-2 is true

Solution

ⁿP_n = n!;

But number of function from A to B is nⁿ

The correct answer is: Both Statements-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1

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Re-Attempt Weekly Quiz Competition

Permutation Test - 11

Result

Permutation Test - 11

Score

-

Rank

-

SHARING IS CARING

Question 1

None of these

n

2

3

Solution

Question 2

200

160

None of these

80

Solution

Question 3

(n + 2) 2(n–1)

3nCn – 2nCn

3nCn

3n

Solution

Question 4

48

658

976

1048

Solution

Question 5

29 – 2

29 – 1

29 – 4

29

Solution

Question 6

Both Statements-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1

Both Statements-1 and Statement-2 are true and Statement-2 is not the correct explanation of Statement-1

Statement-1 is true and Statement-2 is false

Statement-1 is false and Statement-2 is true

Solution

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(n + 2) 2^(n–1)

³ⁿC_n – ²ⁿC_n

3nC_n

2⁹ – 2

2⁹ – 1

2⁹ – 4

2⁹