Combinatorics review problems.

1. (1) 7 teams compete in a men’s hockey league. If each team plays each other
twice, how many games are necessary to complete the league schedule?
(a) 42 (b) 21 (c) 84 (d) 52

2. (2) The sum of the seventh row of Pascal’s triangle is the same solution
to;
(a) The number of solutions a student could get if they guessed on six
questions of a multiple choice exam, each question having four answers.
(b) The number of solutions a student could get if they guessed on six
questions of a true false exam.
(c) The number of solutions a student could get if they guessed on seven
questions of a multiple choice exam, each question having four answers.
(d) The number of solutions a student could get if they guessed on seven
questions of a true false exam.

3. (3) Suppose the last four digits of a telephone number must include at least one
repeated digit. How many such numbers are there?
(4) A multiple choice exam has 20 questions, each with four possible
answers, and 10 additional questions, each with five possible answers.
How many different answer sheets are possible?

4. (5) In a 52 card deck, how many 5 card poker hands are possible that have
exactly two pair? one pair?

5. (a) How many terms are there in the expansions of (x + y)9 and (x + y)10?
(b) Which of the expansions in part (a) has a middle term?
(c) Under what conditions does the expansion of (x + y)n have a middle
term?

6. There are 9 points marked in a plane, no three of which lie in a straight line.
(a) How many straight lines can be drawn, each containing 2 of the points?
(b) How many of these pass through one or more of 3 specified points in
the set?

7. (a) How many different 4 digit numbers are there in which all the digits
are different?
(b) How many of these numbers are odd?
(c) How many of these numbers are divisable by 5?

8. (a) In how many ways can 4 English books and 3 French books be arranged
in a row on a shelf?
(b) In how many of these ways will the French books be together?
(c) In how many ways can the books be arranged if the English books are all
identical, but the French books are not?

9. (a) How many 3-digit numbers can be formed if no digit is used more than
twice in the same number?
(b) How many of these numbers are odd?
(c) How many of these numbers are divisable by 5?

10. A party of 18 people is divided into 2 different groups consisting of 11
people and 7 people. The number of different ways this can be done is:

11. In the binomial expansion of , the term containing is the:

12. In how many ways can 10 people be seated at a round table?
Leave your answer in factorial notation.

13. Simplify:

14. In the binomial expansion of there is a term that when simplified
contains . Find and simplify this term completely.