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mathematics UK Year 12 Mathematics (AS)

Solved problems for the chapter:

Permutations and Combinations

View the theory »

Combinations

1. Combinations without repetition

2. Combinations with repetition

3. Forum

View the theory »

Fundamental counting principle for KS4

1. Product rule or basic theorem of combination

2. Addition rule

3. Forum

↓ Show theory [2] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:permutations_and_combinations/4764/

Example 1.

Loto game is played by announcing a combination of seven numbers from 1 to 39. Use the information to show how many possible outcomes there are.

Example 2.

There are 10 tulips and 17 chrysanthemums in a bouquet. How many ways can we chose to get the following pairs of flowers?

there are only two tulips in the pair,
there are only two chrysanthemums in the pair,
in a pair, is one tulip and one chrysanthemum,
two flowers are optional in a pair.

What are all the combinations of four digits? Note that the digits of the numbers are from 0 to 9.

7 friends meet in the city. How many times do they shake hands if everyone shakes hands with everyone?

28 students entered the school math competition. The top four then qualify for the regional competition.

How many possible groups of four are there for the regional competition?
How many ways can students take positions from first to fourth?

How many ways can we store 8 different books in 3 drawers if we need to store four books in the largest drawer and two in the smaller two?

Find n to be valid:

Volleyball match is won when one of the teams gets three of the five sets. Use the combination tree to show all the possibilities at the end of the match.

A family has three children. In how many ways can parents distribute 21 gifts to children so that each receives 7 gifts?

10.

Find n to be valid:

11.

We have 8 daisies and 4 roses available. How many ways can we compose a bouquet of six flowers if there must be at least one rose in the bouquet?

12.

Find n to be valid:

13.

We have 6 letters at our disposal: A, M, R, O, S, I. How many words can we compose if we know that the word ends with a vowel and contains all six letters?

14.

Mc Donald pour drinks in three different sized pots (small, medium and large). They have 4 types of drinks available (coca-cola, fanta, sprite and orange juice). How many different ways can you order a drink?

15.

We have circles and crosses at our disposal. We want to put together characters with a maximum of five circles and/or crosses. How many such characters can we compose?

16.

With the letters of the word SUNCKA we compose words of length 4 without repeating the letters.

How many words:

can be assembled,
can be composed of only consonants,
begins with a vowel,
begins and ends with a consonant,
begins with the letter C,
contains the letter K?

17.

In three different stores they label items. Each designation consists of five characters which comprises digits and letters of the English alphabet. Digits are considered to be repeatable in all cases. The item code is different in each store:

in the first store the label consists of two letters (they must not be the same) followed by three digits,
In another store, the label consists of one letter, followed by three digits and finally another letter (the letters can be the same),
in the third store, however, the label consists of two digits and is followed by three letters, which must not be the same.

In which store can they label the most items?

18.

The company has 7 men and 3 women. How many ways can they be divided into three offices if there can be 4 people in one office and 3 people in the other two, and if the following applies:

3 women are in one office,
no special requirements, how they are distributed among offices,
There should be 2 men and 2 women in the office for 4 people.

19.

Calculate:

20.

There are 20 students in the class. Each of them is dedicated to one sport activity:

10 students are runners,
7 are cyclists
in 3 are swimmers.

We bring three students together in a group.

How many different groups of three students can we put together?
How many different groups of three runners can we put together?
In how many selections is each sport represented?
How many groups of three can we have if there will be at least one rider in each group?

21.

From a set of beads, which includes yellow, green, red and blue beads, draw two beads. Write down all possible color combinations (we do not care about the order of the drawn balls).

22.

We have 5 clothes containers. In how many ways can we select three clothes if:

We do not set limits (clothes can be repeated)?
No clothes is to be repeated?

23.

We have 21 singers available for the composition of the singing quartet, of which two are bassists. How many quartets can we put together if we don't want to have two identical bassists in a quartet?

24.

A bakery has prepared a special package of seven bakery products. The package can include buns, popcorn and cheese loaves. How many different packages did the bakery have to prepare in order for the first customer to choose their favorite combination?

25.

Show that the following equality holds for as a property of Pascal triangle:

26.

In a prize draw, there are 10 red and 20 blue tickets in the bag. We pull four leaflets blindly from the bag. The prize is won if there are no more than two red ones between the tickets. How many ways can we win a prize?

27.

Alex, Bettty, Maria, Nina and Thiago randomly line up for tickets. What is the probability of the event A that the girls are standing at the beginning of the line?

28.

The seller has 15 tickets in the bag: 5 winning and 10 empty. We buy 4 randomly selected tickets. Calculate the probability of events:

A - at least one purchased lottery ticket will win,
B - two purchased tickets will be winning, two will not.

29.

The teacher will randomly divide tests of two types among 12 students (among them Drake, Jones and Milan, others have different names): 6 students will write a test A, 6 student test B. What is the probability that all three (Drago, Jaka and Milan) will write the test A? What is the probability that at least one of the three will write the test B?

30.

From the group of boys and girls we randomly select persons. Calculate the probability of the event that three boys and one girl will be selected.

31.

The English alphabet has one letter more than the Slovenian. We form 5 letter words from both alphabets. What is the difference between the numbers of possible formation of:

words of the English and Slovenian alphabets,
words containing 3 vowels and 2 consonants?
words containing 3 different vowels and 2 different vowels

32.

We randomly choose three gifts from five books, three toys and two packets of candy. Calculate the probability of an event that we selected one book, one toy, and one pack of candies.