1. In how many ways can 11 math books and 5 different physics books be arranged on a shelf if the physics books are to be together and if 4 of the math books are identical and the rest of the math books are non-identical?
  2. In how many ways can the letters of the word myworld be arranged

(a)   if no letter is to keep its original position?

(b)   if the restriction in (a) is removed?

  1. (a) Find the coefficient of x-4 in the binomial expansion of (x3-(2/x5))20

(b) Find the middle term in the binomial expansion of (x3-(2/x5))20.

  1. A 12-member committee has to be chosen from a group of 11 mathematicians, 15 computer scientists and 7 engineers. Dr. R is one of the 11 mathematicians. The committee has to include exactly 5 computer scientists and Dr. R. In how many ways such a committee can be formed?
  2.  In how many ways can 9 red balls and 6 blue balls be placed in 30 different boxes with:

(a)   At most one ball to a box?

(b)   No limit on the number of balls in each box?

  1. In a group of 40 people, 23 know English, 15 know French, and 7 know both French and English

(a)   How many people speak only French?

(b)   How many people speak neither English nor French?

(c)   How many people speak exactly one of the languages English and French?

  1. 200 people are to be distributed into 17 rooms. Each room has 20 chairs.

(a)   Prove that one of the rooms will have at least 12 people.

(b)   Prove that one of the rooms will have at least 9 empty chairs.

  1.       (a)  How many 7-digit numbers can be formed if repetition is not allowed?

(b)   How many 7-digit numbers have 1 or more repeated digits?

(c)   How many odd 7-digit numbers have no repeated digits?

(d)   How many even 7-digit numbers have no repeated digits and do not include the digit 2?