Problems on Counting

In exercises 1 - 5, two dice are rolled one blue and one red.

  1. How many outcomes are possible?
  1. How many outcomes are doubles (both dice show the same number)?
  1. How many outcomes have the blue die showing 2?
  1. How many outcomes have at least one die showing 2?
  1. How many outcomes have neither die showing 2?

Exercises 6 – 10 refer to 8-bit strings

  1. How many 8-bit strings begin 1100?
  1. How many 8-bit strings begin and end with 1?
  1. How many 8-bit strings have either the second or the forth bit or both equal to 1?

There are 3 choices for second and the fourth bit: 01, 10, 11

For each choice there are 26 possible assignments of values to the remaining 6 bits.

Thus the total number of outcomes is 3x26

  1. How many 8-bit strings have exactly two 1's?
  1. How many 8-bit strings have at least one 1?

In exercises 11 - 14, a six-person committee composed of Alice, Ben, Connie, Dolph, Egbert and Frank is to select a chairperson, secretary and treasurer.

  1. How many selections exclude Connie?
  1. How many selections are there in which neither Ben nor Frank is an officer?
  1. How many selections are there in which Dolph is an officer and Frank is not an officer?
  1. How many selections are there in which Ben is either a chairperson or treasurer?

In exercises 15 - 20, the letters ABCDE are to be used to form strings of length 3

  1. How many strings can be formed if we allow repetitions?
  1. How many strings can be formed if we do not allow repetitions?
  1. How many strings begin with A, allowing repetitions?
  1. How many strings begin with A, not allowing repetitions?
  1. How many strings contain A, allowing repetitions?
  1. How many strings contain A, not allowing repetitions?

In exercises 21 - 23, determine how many strings can be formed by ordering the letters ABCDE subject to the conditions given

  1. Containing the substring ACE.
  1. Containing the letters ACE in any order.
  1. Containing either the substring AE or the substring EA
  1. In some versions of FORTRAN, an identifier consists of a string of one to 6 alphanumeric characters beginning with a letter (an alphanumeric character is one of A to Z or 0 to 9.) How many valid FORTRAN identifiers are there?

1