How Many Strings Can Be Formed if We Allow Repetitions Variation of One of Your Values Exam please be careful while you do the work the values are given on top. please consult with me before you start as well. COMPLETE THIS SECTION FIRST:
I.
Before completing the assignment, you will first calculate a few values. These values will be
used throughout the assignment to “randomize” questions. Clearly write your values for a,
b and c at the top of your assignment.
a. [ a=3
II.
b=19 c= 12]
A randomized question might look something like: “the letters ABCDE are used to form strings
of length [c]. How many strings can be formed if we allow repetitions?” So, for me the question
would be: “the letters ABCDE are used to form strings of length 5. How many strings can be
formed if we allow repetitions?” Your question may be slightly different.
III. Before solving, write your version of the question out.
IV. You may need to find a variation of one of your values. For example, since my value of c is 5, I
would replace [c+2] with 6.
Please note that the value of a is
3, b is 19 and c is 12.
QUESTIONS:
1. (12 points) Let the sequence (xn) be defined by x1 = [a], x2 = [b], xn = xn−1 + xn−2 for n ≥ 3. a. Find x3,
x4, x5
b. Find
c. Find
2. (10 points) Find the solution to the recurrence relation an = 3an−1 +10an−2 with initial terms a0 = [a] and
a1 = [b].
3. (5 points) Let the universal set be U = {0,1,2,3,4,5,6,7,8,9,10,11}. Express {[a],[a + 1],[a + 2]}∪
{1,3,6,10} as a bit string.
4. (6 points) Find the coefficient of x5 in the expansion of (x + 2)[c+5].
5. (15 points) A [a+5]-person committee is to select a chairperson, secretary, and treasurer. Nobody can
hold more than one of these positions.
a. Connie and Dolph are both members of the committee. How many selections exclude both Connie
and Dolph?
b. Dolph is a member of the committee. How many selections are there in which Dolph is either a
chairperson or he is not an officer?
c. Ben and Alice are members of the committee. How many selections are there in which either Ben is
chairperson or Alice is secretary?
6. (6 points) The letters ABCDEF and numerals 1, 2, 3, 4 , 5, 6 are to be used to make passwords of
length [a+2]. How many passwords can be formed if we allow repetitions?
7. (6 points) The letters ABCDEF and numerals 1, 2, 3, 4, 5, 6 are to be used to make passwords of length
[a+2]. How many passwords can be formed if we do not allow repetitions?
8. (6 points) In a certain lottery you select [a+5] distinct numbers from 1 through 39, where order makes
no difference. How many different ways can you make your selection?
9. (6 points) A handful of jellybeans is drawn from a jar that contains 5 different flavors: blueberry,
popcorn, pineapple, apple, lemon. How many ways are there to select a handful of [b+6] jellybeans
from the jar?
10. (6 points) How many [c+7]-bit strings either begin with 00 or end with 111?
11. (6 points) What is the minimum number of students, each of whom comes from one of the 50 states,
who must be enrolled in a university to guarantee that there are at least [10b] who come from the same
state?
12. (6 points) Find the number of rearrangements of your first name. Clearly write your first name with
this problem and show your work! [ First name is PUJAN]
13. (10 points) Use induction to prove:
Proposition. For each natural number n, ([c + 3]n − 1) is a multiple of [c + 2].
COMPLETE THIS SECTION FIRST:
I.
Before completing the assignment, you will first calculate a few values. These values will be
used throughout the assignment to “randomize” questions. Clearly write your values for a,
b and c at the top of your assignment.
a. [ a=3
II.
b=19 c= 12]
A randomized question might look something like: “the letters ABCDE are used to form strings
of length [c]. How many strings can be formed if we allow repetitions?” So, for me the question
would be: “the letters ABCDE are used to form strings of length 5. How many strings can be
formed if we allow repetitions?” Your question may be slightly different.
III. Before solving, write your version of the question out.
IV. You may need to find a variation of one of your values. For example, since my value of c is 12,
I would replace [12+2] with 14.
Please note that the value of a is
3, b is 19 and c is 12.
QUESTIONS:
1. (12 points) Let the sequence (xn) be defined by x1 = [a], x2 = [b], xn = xn−1 + xn−2 for n ≥ 3. a. Find x3,
x4, x5
b. Find
c. Find
2. (10 points) Find the solution to the recurrence relation an = 3an−1 +10an−2 with initial terms a0 = [a] and
a1 = [b].
3. (5 points) Let the universal set be U = {0,1,2,3,4,5,6,7,8,9,10,11}. Express {[a],[a + 1],[a + 2]}∪
{1,3,6,10} as a bit string.
4. (6 points) Find the coefficient of x5 in the expansion of (x + 2)[c+5].
5. (15 points) A [a+5]-person committee is to select a chairperson, secretary, and treasurer. Nobody can
hold more than one of these positions.
a. Connie and Dolph are both members of the committee. How many selections exclude both Connie
and Dolph?
b. Dolph is a member of the committee. How many selections are there in which Dolph is either a
chairperson or he is not an officer?
c. Ben and Alice are members of the committee. How many selections are there in which either Ben is
chairperson or Alice is secretary?
6. (6 points) The letters ABCDEF and numerals 1, 2, 3, 4 , 5, 6 are to be used to make passwords of
length [a+2]. How many passwords can be formed if we allow repetitions?
7. (6 points) The letters ABCDEF and numerals 1, 2, 3, 4, 5, 6 are to be used to make passwords of length
[a+2]. How many passwords can be formed if we do not allow repetitions?
8. (6 points) In a certain lottery you select [a+5] distinct numbers from 1 through 39, where order makes
no difference. How many different ways can you make your selection?
9. (6 points) A handful of jellybeans is drawn from a jar that contains 5 different flavors: blueberry,
popcorn, pineapple, apple, lemon. How many ways are there to select a handful of [b+6] jellybeans
from the jar?
10. (6 points) How many [c+7]-bit strings either begin with 00 or end with 111?
11. (6 points) What is the minimum number of students, each of whom comes from one of the 50 states,
who must be enrolled in a university to guarantee that there are at least [10b] who come from the same
state?
12. (6 points) Find the number of rearrangements of your first name. Clearly write your first name with
this problem and show your work! [ First name is PUJAN]
13. (10 points) Use induction to prove:
Proposition. For each natural number n, ([c + 3]n − 1) is a multiple of [c + 2].
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