Discrete Mathematics Answers

Basic Counting Principle

6.     How many different car license plates can be constructed if the licenses contain three letters followed by two digits if:

a.) Repetitions are allowed;

b.) repetitions are not allowed.

7.     Two dice are rolled, one blue and one red. How many outcomes have either the blue die 3 or an even sum or both?

8.     How many integers from 1 to 10,000, inclusive, are multiples of 5 or 7 or both?

9.     Prove that if five cards are chosen from an ordinary 52-card deck, at least two cards are of the same suit.

10. Eighteen persons have first names Adrian, Jheo and Ghimel and last names Ablir and Testor. Show that at least three persons have the same first and last names.

1.     Three departmental committees have 6, 12, and 9 members with no overlapping membership. In how many ways can these committees send one member to meet with the president?

2.     Two dice are rolled, one blue and one red. How many outcomes give the sum of 5 or the sum of 9?

3.     The options available on a particular model of a car are five interior colors, six exterior colors, two types of seats, three types of engines, and three types of radios. How many different possibilities are available to the consumer?

4.     Two dice are rolled, one blue and one red. How many outcomes are possible?

5.     How many times is How many eight-bit strings read the same from either end? (An example of such an eight-bit string is 01111110. Such strings are called palindromes.)

Let R and S be relations on X. Determine whether each statement is true or false. If the statement is true, prove it; otherwise, give a counterexample.

1.     If R is transitive, then R⁻¹ is transitive.

2.     If R and S are reflexive, then R ◦ S is reflexive

3.     If R and S are symmetric, then R ∩ S is symmetric.

4.     If R and S are antisymmetric, then R ∪ S is antisymmetric.

5.     If R is antisymmetric, then R⁻¹ is antisymmetric.

Produce a truth table for given Boolean expression (A+B'+C)(A+B+C)(A'+B+C')

In a school 844 students have access to three software packages A, B, C.

Where 743 didn’t use any software, 740 used only package C, 742 used only package

A,741 used package B, 739 used all three packages, 738 used both A and C

a) Draw a Venn diagram with all sets enumerated as for as possible.

b) If twice as many students used package A as Package C, write down a pair

of simultaneous equations in x and y.

c) Solve these equation to find x and y.

d) How many students used package B.

(b) For any natural number n, prove the validity of given series by mathematical induction:

In a school, n+100 students have access to three software packages A, B and C.n-1 did not use any software  ,  n-2 used only packages A  n-3 used only packages B     ,   n-4 used only packages C

      n-5 used all three packages,n-6 used both A and B.where n is your arid number i.e. 19-arid-234 take n=234 a)    Draw a Venn diagram with all sets enumerated as for as possible. Label the two subsets which cannot be enumerated as x and y in any order.

b)    If twice as many students used package B as package A, write down a pair of simultaneous equations in x and y.c)Solve the equations to find x and y. d)How many students used package C?