Examples and Hints to Homework
 #21  27 page 171

How many eightbit strings begin 1100 ?
Answer : 2^{4} .since there are only four bits to choose.
 How many 8bit strings begin and end with 1?
Answer : 2^{6} since first and last bit have been already determined.
 How many 8bit strings have either the second or the fourth bit 1 (or both)?
Answer: 2^{7} + 2^{7}  2^{6} ( # of 8bit strings with second bit 1 plus
the # of 8bit strings with fourth bit 1 minus the # of 8bit strings
with both second and fourth bit 1 )
or
2^{8}  2^{6} (total number of 8bit strings minus the # of
8bit strings with both second and fourth bit 0)
 How many 8bit strings have exactly one 1?
Answer : 8 = C(8,1)
 How many 8bit strings have exactly two 1's ?
Answer : C(8,2) = 8!/(2!6!)
 How many 8bit strings have at least one 1?
Answer = 2^{8}  1 = total # of 8bit strings minus the # of 8bit strings with no 1's
 How many 8bit 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 8bits string uniquely determine the string! So, how many
4bit 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 : 5^{3} .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?
Answer : 5^{2} .
 How many strings begin with A if repetitions are not allowed?
Answer: 4 * 3
 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?
Answer : #34 minus #38
 How manu strings contain letter A if repetitions are not allowed?
Answer : #35 minus #39
 #42 52 :integers from 5 to 200 , inclusive,
 How many numbers are there ?
Answr = 200  (51)
 How many are even ?
Answer : half of them
 How many are divisible by 5 ?
Answer : [200/5]
 How many contain the digit 7 ?
Answer : singledigit 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 (2digit 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 (signledigit case) + (90  99) /2 (2digit case) +
(3digit case : 1XY ) (100  10)/2
 how many consist of distinct digits?
Answer: (singledigit case ) 5 +
??? (doubledigit case ) + (threedigit 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!
 #3136 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?
Answer = C(6+7,5)
 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+ 71, 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?
Answer = C(7,4) + C(6,1)*C(7,3)
 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?
Answer = C(13,4)  C(11,2)
 How many 8bit strings contain exactly three 0's ?
Answer = 8!/(3!5!) or C(8,3)
 How many 8bit strings contains three 0's in a row and five 1's?
Answer = 6!/5!
 #63  66 refer to a shipment of 50 microprocessors of which four are defective.
 In how many ways can we select a set of four microprocessors?
answer = C(50,4)
 In how many ways can we select a set of four nondefective microprocessors?
Answer = C(504,4)
 In how many ways can we select a set of four microprocessors containing
exactly two defective microprocessors?
Hint : How many ways to select 2 defectives and how many ways to select two nondefective?
 In how many ways to select a set of four containing at least one
defective?
Answer = C(50,4)  C(504,4)