Exercise 3. There are seven coats in a coat check. (i) How many ways can they be

Question

Exercise 3. There are seven coats in a coat check. (i) How many ways can they
be returned to eight people if each person gets at most one coat back and all seven coats are
returned?

Exercise 4. How many ways can 3 pieces of candy be given to 5 people? A single person may
get all three pieces of candy.

Exercise 5. How many ways can a team of 5 people be chosen from a group of 9 if the order
in which the people are chosen matters?

Exercise 6. Suppose that 3C students receive a list of 10 questions that may end up on the
rst midterm. Suppose that four of these questions are randomly chosen to be on the midterm
and there are only four questions on the midterm. How many dierent midterm exams can be
written?

Exercise 7. How many distinct words (they need not make sense) can you make using all the
letters from the word Tennessee?

Exercise 8. Boudica, Syagrius, Vercingetorix, Attila, Clovis, and Theodoric all sit down at
a round table to grab a cup of coee. Clovis and Syagrius do not like each other and will
not sit side by side. (i) How many ways can the 6 people sit at the table if the chairs are
all indistinguishable? (ii) How many ways can the 6 people sit at the table if the chairs are
distinguishable?

Exercise 9. Human cells have twenty three pairs of chromosomes. One chromosome from each
pair is randomly selected to form a parent's contribution of their genetic material to their child.
How many ways may this selection be made?

Exercise 10. In a certain experiment, three drugs are to be tested on a group of thirty indi-
viduals. Each individual is randomly given one of the three drugs. In how many ways can the
experiment be performed?
1

Exercise 11. In a certain experiment, one person from a group of thirty will be selected to
take one of three drugs. In how many ways can the experiment be performed?

Exercise 12. In a certain experiment, three drugs will be given to three dierent people chosen
randomly from a group of thirty. In how many ways can the experiment be performed?

Exercise 13. In a certain experiment, three drugs will be given to three people chosen randomly
from a group of thirty. It is possible that a single person gets more than one drug. In how
many ways can the experiment be performed?

Exercise 14. The seven top scoring students, one of the three T.A.s, and I line up to take a
picture. If the T.A. and I cannot stand next to each other, how many distinct pictures can be
taken. Note that the order of the students matters as well as which of the 3 T.A.s is chosen.

Exercise 15. How many ways can you form two groups, one of 5 people and the other of 4
people, from a group of 20 people given that no person may be in both groups simultaneously?

Exercise 16. There are twenty students in a class. Four are selected to be a part of the math
club and four are selected to be in the poetry club. It is possible that individuals are selected
to be in both clubs. How many selections are possible?

Exercise 17. There are twenty students in a class. Eight will be selected to get funding from
one of two grants. Each grant will support four people but the people cannot tell which grant
supports them, however, people supported by the same grant must work together. A person
can be given at most one source of support. How many dierent combinations of people may
be chosen to be supported.

Exercise 18. There are twenty students in a class. Eight will be selected to get funding from
one of two grants. , however, people supported by the same grant must work together. Students
can be given two sources of support. How many dierent combinations of people may be chosen
to work together.

Exercise 19. You are to pick 20
owers from a garden for a bouquet. There are 6 kinds of

owers in the garden and more than 20 of each kind. All
owers of the same kind are considered
to be the same, for instance, all roses are considered indistinguishable. If you must have at
least 1
ower of each kind in the bouquet, how many different bouquets can you make?

Exercise 20. Suppose two of the people, Don Quixote and Sancho Panza, are identical twins
wearing exactly the same clothing and therefore cannot be distinguished in the photo, i.e. their
order in the photo does not matter. (i) How many distinct photos can be taken with the
eight people in this case? (ii) Suppose now that Don Quixote and Sancho Panza are wearing
dierent hats and therefore they are distinguishable but they insist on standing together in the
photograph. In this case, how many distinct photos may be taken?
2

Exercise 21. A flea is sitting on the grass near five dogs: Jim, Meg, Al, Sue, and Joukahainen.
The
ea jumps from the ground onto one of the dogs and proceeds to jump from dog to dog,
viciously biting each dog one time. (i) The
ea bites exactly one of the dogs exactly twice
however the
ea does not jump on the same dog twice in a row. In how many ways can the

ea bite the dogs? (ii) Suppose that now the
ea MUST make consecutive jumps (that is jump
twice in a row) on the dog that will be bitten twice. In how many ways can the
ea bite the
dogs?
3

Explanation / Answer

3 8c7 8!/7!(8-7)! 8

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