math 239 assignment 1
MATH 239 Assignment 1
• This assignment is due on Friday, January 18th, 2013, at 10am in the drop boxes outside MC 4067.
Late assignments will not be graded.
• You may collaborate with other students in the class, provided that you list your collaborators.
However, you MUST write up your solutions individually. Copying from another student (or any other source) constitutes cheating and is strictly forbidden.
Exercise 1 (10 pts).
(a) In how may ways is it possible to rearrange the letters in “MISSISSAUGA”?
Hint: Consider in how many ways the letter “M” can be placed, and then in how many ways the letters “I” can be placed, etc
(b) Consider a string using only letters A1 , . . . , An . Let S be the family of strings with exactly i1 letters
A1 ; i2 letters A2 ; . . . ; in letters An (where i1 , . . . , in are some non-negative integers). Find a formula for the cardinality of S.
Hint: This generalizes the problem in part (a), the proof is similar.
Solution:
Give 5 points for each parts. Note, for both (a) and (b) two possible proofs are possible (see alternate proof in part (b)). It suffices for students to give the solution in terms of either products of binomials or factorials.
It is fine for students to prove (b) first and apply it to get (a).
(a) Consider how many choices we have for the positions of each of the letters,
(1) there are
(2) there are
(3) there are
(4) there are
(5) there are
(6) there are
11
1
10
2
8
4
4
2
2
1
1
1
choices for the letter M, choices for the letter I, choices for the letter S, choices for the letter A, choices for the letter U, choices for the letter G,
where in (1), 11 is the total number of letters in “MISSISSAUGA”, 1 is the number of letters M; in (2), 10 is the number of remaining letters to be selected after choosing the letter M, and 2 is the number of letters
I; in (3), 8 is the number of remaining letters to be selected after choosing the letters M, I, and 4 is the number of
• This assignment is due on Friday, January 18th, 2013, at 10am in the drop boxes outside MC 4067.
Late assignments will not be graded.
• You may collaborate with other students in the class, provided that you list your collaborators.
However, you MUST write up your solutions individually. Copying from another student (or any other source) constitutes cheating and is strictly forbidden.
Exercise 1 (10 pts).
(a) In how may ways is it possible to rearrange the letters in “MISSISSAUGA”?
Hint: Consider in how many ways the letter “M” can be placed, and then in how many ways the letters “I” can be placed, etc
(b) Consider a string using only letters A1 , . . . , An . Let S be the family of strings with exactly i1 letters
A1 ; i2 letters A2 ; . . . ; in letters An (where i1 , . . . , in are some non-negative integers). Find a formula for the cardinality of S.
Hint: This generalizes the problem in part (a), the proof is similar.
Solution:
Give 5 points for each parts. Note, for both (a) and (b) two possible proofs are possible (see alternate proof in part (b)). It suffices for students to give the solution in terms of either products of binomials or factorials.
It is fine for students to prove (b) first and apply it to get (a).
(a) Consider how many choices we have for the positions of each of the letters,
(1) there are
(2) there are
(3) there are
(4) there are
(5) there are
(6) there are
11
1
10
2
8
4
4
2
2
1
1
1
choices for the letter M, choices for the letter I, choices for the letter S, choices for the letter A, choices for the letter U, choices for the letter G,
where in (1), 11 is the total number of letters in “MISSISSAUGA”, 1 is the number of letters M; in (2), 10 is the number of remaining letters to be selected after choosing the letter M, and 2 is the number of letters
I; in (3), 8 is the number of remaining letters to be selected after choosing the letters M, I, and 4 is the number of