Exercises 33 42 concern the set of binary strings of length 8 (each character is

Question

Exercises 33 42 concern the set of binary strings of length 8 (each character is either the digit 0 or the digit 1). 33. How many such strings are there? 34. How many begin and end with 0? 35. How many begin or end with 0? 36. How many have 1 as the second digit? 37. How many begin with 111? 38. How many contain exactly one 0? 39. How many begin with 10 or have a 0 as the third digit? 40. How many are palindromes? (See Exercise 11.) 4. How many contain exactly seven Is? 42. How many contain two or more Os?

Explanation / Answer

33. How many such strings:

2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^8 = 256 binary strings.

34. How may bits begin and end with 0:

1 * 2 * 2 * 2 * 2 * 2 * 2 * 1 = 2^6 = 64

35. How may bits begin or end with 0:

Number of bits begin with 0 = 2^7

Number of bits end with 0 2^7

Number of bits begin and end with 0 = 2^6

So 2^7 + 2^7 - 2^6 = 192

36. How many have 1 as the second digit?

2 * 1 * 2 * 2 * 2 * 2 * 2 * 2 = 2^7 = 128

37. How many begin with 111?

1 * 1 * 1 * 2 * 2 * 2 * 2 * 2 = 2^5 = 32

38. How many contain exactly one 0?

There are exactly 8 possible position for 0. So answer will be 8.

39. How many begin with 10 or have a 0 as the third digit?

2^6 begin with 10. 2^7 has third digit 0. 2^5 will be satisfy both.

so 64 + 128 -32 = 160

40. How many are palindromes?

We will have 16 palindromes with binary string of length 8.

2^4 different first halves of the string. We won't have choice for second half.

41. How many contain exactly seven 1's?

There are exactly 8 possible position for 1. So answer will be 8.

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