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mathematics UK Year 12 Mathematics (AS)
 
Solved problems for the chapter:
 

Permutations

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:permutations/4766/


Example 1.

Gorginia bought red rose bulbs, yellow rose bulbs and purple rose bulbs. The bulbs are mixed for her and when she planted them, she didn’t know which ones would grow where. What is the probability that she planted roses of the same color together?

    
Example 2.

How many permutations of order 3 are there if there are 6 different elements available.

    
3.

Matey doesn't know the alphabet yet. What is the probability that he will compose his name when randomly composing words from the letters A, E, Y, M and T?

    
4.

If repetition of digits is not allowed, how many 5-digits numbers can be formed from the digits 1,2,3,4,5,6,7,8 if


  • no additional restrictions,


  • the numbers must be even?

    
5.

How many possible arrangements taking 4 at a time are possible if repetition of elements of the set below are allowed?


    
6.

How many possible arrangements of 5-digit numbers are there, repeating the digits 1 and 2?

    
7.

The numbers 1, 2, 3, 4 and 7 are given.


  • A three-digit number is randomly generated from the given digits, in which the digits can be repeated. What is the probability of forming a number that is divisible by 4?

  • A three-digit number is randomly drawn from the given digits, in which the digits must not be repeated. What is the probability of forming a number that is divisible by 3?

    
8.

Solve:








    

9.

The letters of the name HUBERT are randomly arranged in a straight line, each letter is used exactly once. Calculate the probability of events:


  • A - a compound word starts with the letter T and

  • B - we composed the word TREBUH

    
10.

In a computer game, you can create your own action hero by specifying his clothing, headgear and vehicle. You can choose from five outfits, three headgear and seven vehicles. How many different action heroes can you create?

    
11.

When buying a new car, the buyer can choose between 8 different colors, 3 different interior packages and between diesel and gasoline engines. How many types of car can a buyer choose from?

    
12.

In a World Basketball Championship qualifiers, there are 8 teams in a qualifying group. How many matches do they play in each of the qualifying groups if each of the teams plays two matches on a home and away basis?

    
13.

How many numbers greater than 30000 but less that 50000 can the formed from the digits 1, 2, 3, 4, 5 and 6 if repetition is not allowed.

    
14.

A primary school nutrition organizer has 90 menus from previous years at her disposal. 18 menus for breakfast, 25 menus for morning snack, 27 menus for lunch and 20 menus for afternoon snack. Each week, they prepare a weekly menu that includes all four meals a day. How many different daily menus could be put together?

    
15.

At the boarding station, 20 school children board the train. How many ways can students sit if:


  • there are only 6 vacant seats?


  • there are 30 vacant seats?

    
16.

How many words of 8 letters can be formed from the letters I, J and K if:


  • there is no additional restrictions,


  • there must be four K's and four J's in the word?


    
17.

How many words of 6 letters can be permuted from the letters M, N, O and P if:


  • no further restriction?


  • if O appears exactly three times and M twice?

    
18.

There are three pubs in an area. Two tax inspectors arrive at the scene at the same time. How many ways can they check pubs if the pubs cannot be checked more than once?

    

19.

Eric walks along a meadow where 5 horses graze. He has 3 candies in his backpack. How many ways can he divide them among horses if:


  • there is no restriction,


  • one horse gets a maximum of one candy?

    
20.

Solve:




(Hint: (r) means repetition is allowed)

    
21.

In a chess competition, a total of 56 games were played amongst the contestants, with each contestant competing against another contestant twice. How many chess players took part in the competition?

    
22.

A photo album has 8 pages left unoccupied. How many ways can we add 4 photos to the album if we can add one picture to a page?

    
23.

A company has 6 employees. While two employees have to stay in the office, 4 can go for lunch.


  • How many ways can they be arranged in a row at the canteen?


  • How many such deployments begin with Martha and end with Janez?


  • How many such deployments do not end with Loret and Lukas?

    
24.

How many six-digit numbers:


  • begin with an even number,


  • ends with an odd number,


  • begin and ends with an odd number.


Note that the none of the numbers should start with zero and repetition of digits is allowed.

    
25.

In how many ways can 6 figures be arranged in a row from equal number of white and some black figures, given that a white figure must always occupy the first position?

    
26.

In how many ways can a dog trainer distribute 3 distinct treats among 12 dogs, if:


  • each dog gets a maximum of one treat,

  • one dog can get more treats,

  • each dog gets a maximum of two treats?

    
27.

Find the value of n such that:




(Hint: (r) means repetition is allowed)

    
28.

A room has 10 exit doors. How many ways can 8 people leave the room?

    

29.

The letters A, B, C, D are sorted into strings of length 3. The letters must not be repeated in the string. Calculate how many different sets we can get this way. Write down those among them that begin with B.

    
30.

How many unique 3-digit alarm codes can be created using digits 0-9, with the condition that the third digit must be 9?

    
31.

How many unique three-digit numbers can be formed using the digits 1, 2, 3, 4, 7, and 9, with no repeated digits if


  • no further restriction?


  • the numbers are odd?


  • the numbers are greater than and lesser than ?