Concordia UniversityDepartment of Electrical and Computer EngineeringCOEN231, Introduction to Discrete Mathematics

November 27, 2011

Assignment 5Due Monday December 5

1. Consider bit strings of length 12.

(a) How many begin with 110?

(b) How many begin with 11 and end with 10?

(c) How many begin with 11 or end with 10?

2. (a) A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random withoutlooking at them. How many balls must she select to be sure of having at least three ballsof the same color?

(b) How many license plates can be made using either three letters followed by three digits orfour letters followed by two digits?

(c) How many different strings can be made from the letters in MISSISSIPPI, using all theletters.

(d) How many bit strings of length seven either begin with two 0s or end with three 1s?

3. (a) What is the coefficient of x8y9 in the expansion of (3x + 2y)17?

(b) What is the coefficient of x101y99 in the expansion of (2x− 3y)200?

4. (a) In how many ways can forty identical (indistinguishable) objects be distributed in fourdistinguishable boxes, if no constraints are placed on the number of objects that can beput in a box?

(b) In how many ways can forty different (distinguishable) objects be distributed in four dis-tinguishable boxes, if exactly ten objects must be placed in each box?

5. (a) Prove the following:(2n2

)= 2(n2

)+ n2

(b) Prove the following identity:n∑

k=1

(n

k

)= 2n

(c) Use mathematical induction and Pascal’s identity to prove(n

0

)−(n

1

)+

(n

2

)− · · · + (−1)k

(n

k

)= (−1)k

(n− 1

k

)(d) Use the previous to prove (

n

0

)+

(n

2

)+

(n

4

)+ · · · = 2n−1

and (n

1

)+

(n

3

)+

(n

5

)+ · · · = 2n−1

1