C960 Discrete Mathematics II

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Access and unlock Multiple Practice Question for C960 Discrete Mathematics II to help you Pass at ease.

Free C960 Discrete Mathematics II Questions

1.

How many vertices does the complete 4-ary tree of height 3 have?

  • 21

  • 40

  • 85

  • 88

Explanation


2.

What is the number of onto functions from a set of 10 elements to a set of 3 elements?

  • 59049

  • 19683

  • 270

  • 27

Explanation

The number of onto functions from a set of size mmm to a set of size nnn can be calculated using inclusion-exclusion:

Number of onto functions=nm-n1n-1m+n2n-2m

 


3.

Using fast exponentiation, compute 5⁴⁰ mod 17.

  • 1

  • 2

  • 4

  • 8

Explanation

By Fermat: 5¹⁶ ≡ 1 mod 17.

40 = 2×16 + 8 → 5⁴⁰ = 5^{32+8} = (5¹⁶)² × 5⁸ ≡ 1² × 5⁸ mod 17.

Now compute 5⁸:

5¹ = 5

5² = 25 ≡ 8

5⁴ = 8² = 64 ≡ 13

5⁸ = 13² = 169 ≡ 169−9×17 = 169−153 = 16 ≡ −1 mod 17

So 5⁴⁰ ≡ −1 ≡ 16 mod 17, 1 

5⁴⁰ = (5¹⁶)² × 5⁸ ≡ 1 × (−1) = −1 ≡ 16


4.

Find X if 7x≡2mod13(discrete logarithm problem).

  • 3

  • 5

  • 7

  • 11

Explanation


5.

Using Fermat’s Little Theorem, compute 3100mod 7.

  • 1

  • 2

  • 3

  • 4

Explanation

Fermat’s Little Theorem states that if ppp is a prime number and aaa is an integer not divisible by p, then ap-1−1≡1(mod p).


6.

What is the result of 47 mod 13?

  • 0

  • 5

  • 8

  • 11

Explanation

The modulo operation finds the remainder when one number is divided by another. Here, we divide 47 by 13. First, determine how many times 13 fits into 47: 13×3=39, which is the largest multiple of 13 less than 47. Then, subtract this from 47 to find the remainder: 47−39=8.Therefore, 47 mod 13=8.


7.

How many comparisons does Merge Sort make in the worst case for n=8 elements?

  • 8

  • 12

  • 17

  • 20

     

Explanation

Explanation:

Merge Sort is a divide-and-conquer algorithm that splits the array into two halves, recursively sorts each half, and then merges the sorted halves.

The worst-case number of comparisons can be computed using the recursive formula for comparisons C(n):



where n−1 comparisons are made during the merge step.

For n=8n

1.Split 8 into two halves of 4:C(8)=C(4)+C(4)+7 

2.For C(4): split into 2 + 2:C(4)=C(2)+C(2)+3

3.For C(2): split into 1 + 1:C(2)=C(1)+C(1)+1=0+0+1=1 

Now compute step by step:

C(4)=1+1+3=5 

C(8)=5+5+7=17


8.

How many handshakes occur when 12 people each shake hands with every other person exactly once?

  • 66

  • 72

  • 132

  • 144

Explanation

When each person shakes hands with every other person exactly once, the total number of handshakes can be calculated using the combination formula n2, which counts the number of ways to choose 2 people out of nnn to shake hands. For 12 people, this is:

122=12.112=66



 


9.

How many ways can 7 people sit around a circular table if two specific people must sit together?

  • 240

  • 360

  • 480

  • 720

Explanation

For circular arrangements, we usually fix one seat to account for rotations, giving (n−1)! arrangements for nnn people.

Here, there are 7 people with two specific people required to sit together. Treat these two as a single "block," reducing the problem to 6 objects (the block + 5 other people). The number of circular arrangements is:

6-1!=5!=120

 


10.

What is the closed-form solution for the recurrence
an=5an-1-6an-2 with a0=0,a1=1? 

  • an=2n-3n

  • an=3n-2n

  • an=6n-5n

  • an=5n-6n

Explanation

This is a linear homogeneous recurrence with constant coefficients. Solve using the characteristic equation:

r2-5r+6=0

factor:

r-2r-3=0⇒r=2,3

 


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