1 How many bit strings of length seven either begin with two 0s or end with thre

Question

1 How many bit strings of length seven either begin with two 0s or end with three 1s? Explain your answer

2 How many positive integers not exceeding 100 are divisible either by 4 or by 6? Explain your answer

3 A WEP key for a WiFi network is a string of either 10, 26, or 58 hexadecimal digits. How many different WEP keys are there? Explain your answer or show your work.

4 Justify (provide reasoning) that if seven integers are selected from the set of first ten positive integers, there must be at least two pairs of these integers with the sum 11.

5 How many ways are there for 10 women and 6 men to stand in a line so that no two men stand next to each other? Explain your answer.

Explanation / Answer

Using Inclusion and Exclusion principle:

1. Start with 2 zeros : 2^5
   End with 3 ones: 2^4

Start with 2 zeros and end with 3 ones: 2^2

Result = 32 + 16 - 4 = 44

2. Divisible by 4 : 100/4 = 25

Divisible by 6 : 100/6 =16

Divisible by lcm(4,6) = 12: 100/12 : 8

result: 25+16-8 = 33

3. each digit can be 16 numbers.

So 10 digit key: 16^10

26 digit key: 16^26

58 digit key: 16^58

4. total pairs that sum to 11 = 5

for every 7 numbers = total pairs = 7*3 = 21

5. ... | .... | .... | .... | .... | ....| .... = 16

Let the men be the separators and each gap has to be of length 1

so remaningn women = 11

x1 + x2 + x3 + x4 + x5 + x6 + x7 = 11

(17!)/(11!)(6!)

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