Counting

Exercise 1

You are tossing 10 coins and counting the number of "tails" coming up

a

How many different outcomes contain 3 "tails"?

b

What is the total number of different outcomes?

c

How many outcomes contain at least 8 "tails"?

d

How many outcomes contains no more than 8 "tails"?

Exercise 2

The management of a sports club consists of $6$ members. When meeting, some of them shake hands on arrival.

a

Draw a grid that can help you to determine all possibilities for someone who wants to shake hands with two persons.

b

How many possibilities does this person have?

c

What is the total number of possibilities?

Exercise 3

A commercial representative is supposed to visit $14$ customers this week. The distance to each customer is about the same. He decides to visit $4$ customers the first day.

a

What is the total number of ways in which he can visit $4$ out of $14$ customers?

b

The second day he visits only two customers, because he needs to do some administration work as well. How many possibilities does he have on the second day?

Exercise 4

The end score of the soccer match Ajax–FC Zwolle was 6–4. The way this score developed is shown in the figure.

a

Write down the development of the score by noting the interim scores.

b

Assuming you only know the final score, in how many ways could the score have evolved?

c

Except the final score (6–4) you also know the score at the half way break (4–1). What is the total number of ways the score can have evolved?

Exercise 5

The figure blow depicts a garden with walking paths and a pond. This garden plan can be schematized into a rectangular grid like the one below. Use this grid to calculate the number of paths without detours from the entrance of the garden to the exit.

Exercise 6

The figure shows the international morse code. Each letter consists of at most $4$ signals; each digit consists of exactly $5$ signals. A signal can be 'short' (denoted by - ) or 'long' (denoted by —).

a

How many different symbols can you make using a morse code of 5 signals?

b

How many symbols can you make using no more than 4 signals?

c

It would also be possible to use two dots and three stripes to denote all digits. Show this by systematically writing down all possibilities.

Exercises