## Examples and Hints to Homework

• #21 - 27 page 171
• How many eight-bit strings begin 1100 ?
Answer : 24 .since there are only four bits to choose.
• How many 8-bit strings begin and end with 1?
Answer : 26 since first and last bit have been already determined.
• How many 8-bit strings have either the second or the fourth bit 1 (or both)?
Answer: 27 + 27 - 26 ( # of 8-bit strings with second bit 1 plus the # of 8-bit strings with fourth bit 1 minus the # of 8-bit strings with both second and fourth bit 1 )
or 28 - 26 (total number of 8-bit strings minus the # of 8-bit strings with both second and fourth bit 0)
• How many 8-bit strings have exactly one 1?
• How many 8-bit strings have exactly two 1's ?
• How many 8-bit strings have at least one 1?
Answer = 28 - 1 = total # of 8-bit strings minus the # of 8-bit strings with no 1's
• How many 8-bit strings read the same from either end?
Answer : Since the strings read the same from either end, this means that the first 4 bits of the 8-bits string uniquely determine the string! So, how many 4-bit strings out there?
• #32 - #33 page 171
• How many selections are there in which either Dolph is chairperson or he is not an officer?
Answer : case that Dolph is the chair plus the case that Dolph is not an officer, and these two cases are mutually disjoint.
• How many selections are there in which Ben is either chairperson or treasurer?
Answer : C(5,2)*2! + C(5,2)*2! or 5*4 + 5*4

• (34 - 41 page 171)the letters ABCDE are to be used to form strings of length 3:
• How many strings can be formed if repetitions are allowed?
Answer : 53 .Since for each of the three positions , we hae five choices.
• Same as before , but repetitions are not allowed.
Answer : 5 * 4 * 3
• How many strings begin with A , allowing repetition?
• How many strings begin with A if repetitions are not allowed?
• How many strings do not contain the letter A, allowing repetitions?
Answer : ?? (similar to #34 but take letters only from BCDE)
• How many strings do not contain the letter A, if repetitions are not allowed?
Answer : ?? (Similar to #35, but takes letters from BCDE).
• How many strings contain letter A, allowing repetition?
• How manu strings contain letter A if repetitions are not allowed?
• #42- 52 :integers from 5 to 200 , inclusive,
• How many numbers are there ?
Answr = 200 - (5-1)
• How many are even ?
• How many are divisible by 5 ?
• How many contain the digit 7 ?
Answer : single-digit case : 1 ; double digit case : 10 + 9 - 1 ; three digit case : (1XY) 10 + 10 -1
• How many do not contain 0 ?
Answer : 5 (single digit case) + 9*9 (2-digit case) + 9*9 (1XY case)
• How many greater than 101 and do not contain the digit 6 ?
Answer : 1 (case 200) + 9*9 (1XY case) - 2 (case 101 and case 100)
• How many have the digits in strictly increasing order?
Answer : 5 (signle-digit case) + (90 - 9-9) /2 (2-digit case) + (3-digit case : 1XY ) (100 - 10)/2
• how many consist of distinct digits?
Answer: (single-digit case ) 5 + ??? (double-digit case ) + (three-digit case : 1XY)
• How many are of the form xyz, where 0 < x < y and y > z ?
Since x must be 1 , then y > 1, so for y between 2 to 9 , z must less than y. Answer = 2 + 3 + ... + 9 = 44
• #10 - 18 p.182 : determine how many strings can be formed by ordering the letters ABCDE subject to the conditions given:
• Contains the substring ACE : 3!
• Contains the letters ACE together in any order : 3! * (3!)
• contains the substrings DB and AE : 3!
• contains either the substring AE or the substring EA : 2 * 4!
• A appears before D : 5!/2
• Contain neither of the substring AB, CD
5! - number of strings contains either AB or CD (or both)
• Contains neither of the substring AB, BE :
5! - { 4! + 4! - 3! }
• A appears before C and C appears before E :
5! / 3!
• Contains either the substring DB or the substring BE :
4! + 4! - 3!
• #31-36 refer to a club consisting od six distinct men and seven distinct women :
• In how many ways can we select a committee of five persons?
• In how many ways can we select a committee of three men and four women?
Hint : use multiplication principle
• In how many ways can we select a committee of four persons that has at least one woman?
Answer = C(7,1) * C(6+ 7-1, 3) or C(6+7,4) - C(6,4)
• In how many ways can we select a committee of four persons that has at most one man?
• In how many ways can we select committee of four persons that has persons of both sexes?
Answer = C(13,4) - C(6,4) - C(7,4)
• In how many ways can we select a committee of four so that Mabel and Ralph do not serve together?