8.05 Permutations (Enrichment)
Evaluate the following expressions:
5!
\dfrac{7!}{5!}
{}^{6}P_{3}
{}^{5}P_{5}
{}^{4}P_{0}
{}^{11}P_{1}
{}^{7}P_{3}
{}^{8}P_{6}
How many ways can 9 different items be arranged in a line?
Five cards have different letters written on them. The letters are M, L, I, E, S. The cards are shuffled and laid out on a table with the letters face up next to one another. How many possible arrangements are there?
How many different ways can the letters of the word 'SHELF' be arranged?
Determine the number of ways the letters of the word 'PASSIONATE' can be arranged in a line.
If there are 5 swimmers in a race, in how many different orders can they finish (assuming there are no ties)?
In how many ways can 4 women and 5 men enter a lifeboat if women are to enter first?
In a certain code, the digits 0 and 1 are placed together to form a string and each string represents a word. For example, 0011010 is a string.
How many of these strings can be created using two 0s and four 1s?
If an mp3 player contains 7 songs, in how many different ways could all 7 songs be played, with no song being repeated?
To determine the successful applicant for a job, the interviewer assigns a problem at random to each applicant. If there are 4 applicants, how many different ways can the problems be assigned among them?
In the qualifying round of a car race, the 8 cars are given random starting positions. In how many different ways can starting positions be allocated?
A bank requires its customers to create a 4-digit pin for their bank card. They can use the digits 0 to 9, with repetitions allowed. How many different bank cards can the bank issue if each card must have a different pin?
An exam consists of 8 multiple choice questions, where the correct answer could be \text{A, B, C} or \text{D} . If the answer to each question is guessed randomly, how many different ways are there of answering all 8 questions?
Katrina's ATM pin is a 4-digit number. She remembers the first digit, but cannot remember the next three digits that follow. How many different possibilities are there?
Christmas stockings are to be hung in a line along a fireplace. There are 2 green, 4 blue and 5 red stockings left over from the previous year. Find how many different arrangements are possible if:
All the stockings are used.
At least 10 stockings are used.
A city has to accommodate for 9.96 million households requiring a telephone number. They are about to issue 8-digit telephone numbers, with 5 as the first digit.
How many different telephone numbers can be created if repetition of digits is not allowed?
How many different telephone numbers can be created if repetition of digits is allowed?
If each household is allowed one telephone number where digits can be repeated, will there be enough telephone numbers for the number of households?
Motorists can create personalised licence plates consisting of 6 characters, and they can be arranged in any way the motorist chooses.
Tina wants to use her three favourite letters \text{L, Y} and \text{X}, and her three favourite digits, 6, 9 and 1.
How many different licence plates can she create if the letters and digits can appear in any particular order (and no character can be repeated)?
How many different licence plates can she create if the three letters are followed by the three digits (and no characters can be repeated)?
At a gathering of political leaders, there are 2 delegates from European countries, 3 delegates from African countries, and 4 delegates from South American countries. During question time, they are to be seated in a row on stage, however the South American delegates need to leave early and so are seated together to the right of the other delegates. In how many ways can the delegates be seated?
A photographer would like to display his books on different countries in his new store. He has 5 different books on Argentina, 3 different books on Bulgaria, and 4 different books on Cambodia.
Find the number of ways these books can be arranged on a shelf if:
They can be in any order.
The countries are in alphabetical order from left to right.
Books of the same country have to be kept together.
At a motor show, one particular car maker wants to arrange its 5 new cars in a display line.
In how many ways can the 5 cars be arranged if each of them is a different model?
If 2 of the cars are the same colour and model, such that they look identical to one another, in how many different arrangements can the 5 cars be displayed if the remaining cars are unique in appearance?