Counting

Exercise 1

Somebody has to answer $10$ questions with 'yes' or 'no'.

a

How many different answer lists with exactly three "yes-es" are possible?

b

How many different answer lists with exactly $9$ "yes-es" are possible?

c

What is the total number of different answer lists?

Exercise 2

You are tossing five different coins and you count the number of "heads".

a

How many different outcomes are possible?

b

How many possible outcomes with exactly two "heads" exist?

c

You now toss $50$ coins. In how many ways can you get $20$ "heads"?

Exercise 3

There are $24$ participants in a chess tournament. They play a single round-robin schedule, so each participant plays every other participant once. The number of games can be calculated using combinations. Explain why this is true and calculate the number of games.

Exercise 4

A group consists of $14$ girls and $12$ boys. A group of four is assigned by lot.

a

If the group should contain only girls, how many groups are possible?

b

Answer the same question if the group should consist of two boys and two girls.

Exercise 5

In how many ways can $8$ different books be placed on a shelf if

a

every order is allowed?

b

the three math books should be placed next to each other?

c

the two dictionaries should be at the end of the row next to each other?

d

three books are picked to be wrapped and then put at the end?

Exercise 6

You are throwing with three dice and you count the number of dots that are on top.

a

How many different outcomes are possible?

b

There are several different ways to throw 12 dots. For instance by throwing three times 4, but also by throwing 6 once and 3 twice.
How many possible ways are there of throwing 12 dots?

Exercise 7

In a comprehensive school the joint decision making council consists of $18$ persons: $9$ employees and $9$ parents and/or students. This council elects an everyday management group of four persons.

a

How many different everyday management groups can be formed if there are no restrictions to its composition?

b

How many groups are possible if the everyday management group should consist of equal numbers of employees and parents and/or students?

c

How many possibilities are there if first the president, then the vice-president, next the secretary and last the treasurer are chosen?

Exercises