Permutation Test

Result

Permutation Test - 8

Score
-
out of -
Rank
-
out of -

TIME Taken - -

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Weekly Quiz Competition

Question 1
1 / -0

The number of triangles whose vertices are at the vertices of an octagon but none of sides happen to come from the sides of the octagon is-

A
24

B
52

C
48

D
16

Solution

Total number of triangles are = ⁸C₃ = 56
Triangle whose one side is side of octagon are
= 8 .(8 – 4) = 32
Triangles whose two sides are sides of octagon are = 4 × 2 = 8
Required number of triangles which none of side not the side of octagon are
= 56 – 32 – 8 = 16
Question 2
1 / -0

A set contains (2n + 1) elements. The number of subsets of the set which contain at most n elements is-

A
2ⁿ

B
2^{n + 1}

C
2^{n – 1}

D
2²ⁿ

Solution

No. of sub sets = ^{(2n + 1)}C₀ + ^{(2n + 1)}C₁ + ^{(2n + 1)}C₂ + ^{(2n + 1)}C₃ + ^{(2n + 1)}C₄+ … + ^{(2n + 1)}C_n
we know,
2^{2n + 1} = ^{(2n + 1)}C₀ + ^{(2n + 1)}C₁ + ^{(2n + 1)}C₂ + … + ^{(2n + 1)}C_n +^{(2n + 1)}C_{(n + 1)} + ^{(2n + 1)}C_{(n + 2)} + … + ^{(2n + 1)}C_2n + ^{(2n + 1)}C_{(n + 1)}
⇒ 2^{(2n + 1)} = 2[^{(2n +} ¹⁾C₀ + ^{(2n +} ¹⁾C₁ + ^{(2n +} ¹⁾C₂ + … + ^{(2n +} ¹⁾C_n]
⇒ 2²ⁿ = ^{(2n +} ¹⁾C₀ + ^{(2n +} ¹⁾C₁ + ^{(2n +} ¹⁾C₂ + … + ^{(2n +} ¹⁾C_n
∴ Option D. is correct answer.