8.06 Probabilities with permutations (Enrichment)

The digits 3, 6, 9, 2, 4 and 5 are used to make numbers that contain 2 or more digits, but the number cannot contain repeated digits. What proportion of all possible numbers are even?

Christa, James, Beth, Buzz and Eileen are to be placed in a line in random order.

The letters of the word SPACE are to be rearranged.

8 cards have different letters written on them. The letters are A, R, I, O, S, C, G, U. The cards are shuffled and laid out on a table with the letters face up next to one another.

A random three-letter word is to be formed from the letters A, B, C, D, E and F without repetition.

A random number is to be formed from the digits 1, 2, 3, 4 and 5 without repetition.

A four digit number is randomly made using each of the digits 1, 5, 4 and 7 once.

The letters of the word ANKLET are randomly arranged in a line.

The letters of the word CINNAMON are randomly arranged in a line.

14 test tubes that contain a different blood sample are to be placed randomly in a circular separator. Two of these tubes belong to twins Susana and Justin. What is the probability that the two test tubes containing their blood samples are placed next to each other in the separator?

In a horse race, a trifecta is the name given to predicting the first three horses in their correct order. In a horse race in which there are 10 horses, what is the probability that Michael picks the trifecta?

In an oral French exam, each of the 9 students will be asked a different question to respond to. Before the exam, they are given 12 questions from which the 9 questions will be chosen. 5 of these 12 questions are related to the environment. Iain, Roald and Rochelle are particularly strong on talking about the environment. What is the probability that the question each of them will be asked is about the environment?

At the opening ceremony of the Commonwealth Games, a country is represented by 13 athletes who are to walk out onto the stadium in a line.

In a children’s memory test, 7 different toys are arranged in a line. The teacher taps 3 of these toys in a random order, and the child has to tap those toys in the same order shown.

In Queensland, standard motor vehicle number plates consist of three digits followed by three letters.

To determine the successful applicant for a job, the interviewer assigns a problem to each of the 6 applicants.

An mp3 player contains 90 songs, of which one is Silent Night. The songs are to be played in a random order (so that each song can be played any number of times). Find the chance that Silent Night will be:

For the multiple choice section of an end of year exam, the examiner noted that there are 5 questions whose answer is \text{A}, 2 questions whose answer is \text{B}, 6 questions whose answer is \text{C} and 7 questions whose answer is \text{D}. To avoid students guessing an answer based on the pattern of answers, she randomly orders the questions. When she looks at the final paper where the multiple choice questions are randomly ordered, she sees that all the first 5 answers are A, the next 2 answers are B, the next 6 answers are C and the last 7 answers are D. What is the probability of this order being randomly generated?

12 delegates from various countries are randomly seated for round table discussions. Two particular delegates have previously had a confrontation and would prefer not to sit next to each other. What is the probability that they are not seated next to each other, nor directly opposite one another?

Three friends run into one another at the airport and discover that they will be on the same flight. The plane they will be flying in consists of 41 rows of seats, where each row has three seats on each side of a central aisle. If all passengers have previously been randomly allocated seats, find the probability that: