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2nd Pass Discrete Mathematics (2011/12) Sem. I
Homework 1
There will be TWO homeworks of which this is the first. The Exam will count for 70 marks
and the two homeworks will count for 30 marks thus totalling 100 marks. IF you score
less than 30% in the exam, your homework marks will be capped at 20 out of
30. This is to prevent diligent copyists getting say 20% in the exam (14 marks from 70)
and say 90% in copying homework (27 marks from 30) and thereby gaining a pass mark of
14 + 27 = 41.
This homework will be collected at the end of the lecture on Wednesday, 2ndd November
2011.
1. In how many ways can a three-letter sequence by formed using the letters u, v, w, x, y, z?
Of these sequences how many contain the letter y but have no letter repeated?
If we allow repetition, how many contain the letter y?
2. How many different non-empty (sub)collections can be formed from a collection of 5
identical black balls and 8 identical white balls?
3. How many numbers in the set {1, 2, . . . , 360} are divisible by (a) at least ONE of the
numbers 6, 8, or 9? (b) precisely TWO of the numbers 6, 8, or 9?
4. The NUI Galway campus telephone numbers consist of 4 digits. How many of the
numbers have one or more digits repeated?
5. How many Poker (5-card) hands consist if a "flush" (i. e. all cards of the same suit)?
How many Poker (5-card) hands have exactly 3 Aces?
6. Dr Quickstep walks using any mixture of short steps of 1 foot or long strides of 3 feet.
Show that if Qs is the number of ways that Dr Quickstep can walk s feet, then (a)
Q1 = Q2 = 1, Q3 = 2 and Qs = Qs-1 + Qs-3 for s 4. Calculate Q12.
7. In how many ways can one distribute 10 e1 coins to six students so that no student
receives more than e2?
8. Miss B. Haviour and Miss D. Meanour played a chess match in which there were no
drawn games. The first player to win three games in a row or a total of five games
won the match. Miss B. Haviour won the first game and the person who won the
second game also won the third game. By constructing an appropriate tree diagram
or otherwise, determine the number of ways in which the match may have proceeded.
Homework 1
There will be TWO homeworks of which this is the first. The Exam will count for 70 marks
and the two homeworks will count for 30 marks thus totalling 100 marks. IF you score
less than 30% in the exam, your homework marks will be capped at 20 out of
30. This is to prevent diligent copyists getting say 20% in the exam (14 marks from 70)
and say 90% in copying homework (27 marks from 30) and thereby gaining a pass mark of
14 + 27 = 41.
This homework will be collected at the end of the lecture on Wednesday, 2ndd November
2011.
1. In how many ways can a three-letter sequence by formed using the letters u, v, w, x, y, z?
Of these sequences how many contain the letter y but have no letter repeated?
If we allow repetition, how many contain the letter y?
2. How many different non-empty (sub)collections can be formed from a collection of 5
identical black balls and 8 identical white balls?
3. How many numbers in the set {1, 2, . . . , 360} are divisible by (a) at least ONE of the
numbers 6, 8, or 9? (b) precisely TWO of the numbers 6, 8, or 9?
4. The NUI Galway campus telephone numbers consist of 4 digits. How many of the
numbers have one or more digits repeated?
5. How many Poker (5-card) hands consist if a "flush" (i. e. all cards of the same suit)?
How many Poker (5-card) hands have exactly 3 Aces?
6. Dr Quickstep walks using any mixture of short steps of 1 foot or long strides of 3 feet.
Show that if Qs is the number of ways that Dr Quickstep can walk s feet, then (a)
Q1 = Q2 = 1, Q3 = 2 and Qs = Qs-1 + Qs-3 for s 4. Calculate Q12.
7. In how many ways can one distribute 10 e1 coins to six students so that no student
receives more than e2?
8. Miss B. Haviour and Miss D. Meanour played a chess match in which there were no
drawn games. The first player to win three games in a row or a total of five games
won the match. Miss B. Haviour won the first game and the person who won the
second game also won the third game. By constructing an appropriate tree diagram
or otherwise, determine the number of ways in which the match may have proceeded.
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