Some Permutations Qs Cambridge (1 Viewer)

[hermine]

Have been annoying me for a while, appreciate the help

1) For the word UNIFORM, how many arrangements are possible if M is somewhere to the right of U?

2) Five Backpackersarrive in a city with five youth hostels. Suppose two backpackers are brother and sister, so wish to stay in the same hostel. How many different accomodation arrangements are there if if the other three can go to any of the other hotels?

3) Numbers less than 4000 are formed from the digits 1, 3, 5, 8 and 9. How many numbers possible?







Answers
1) 2520
2) 320
3) 133

 

[lolokay]

1) how many arrangements of uniform are there? what's the probability that the M is to the right of the U?

2) keep in mind that the other backpackers can go to the same hostel if they want, just not the one with the brother/sister

3) keep in mind each digit can only be used once, and the number can use any number of those digits (ie. between 1 and 4). Just take it by cases: if there are 4 digits, how many numbers can go in the first spot? then the second? etc.


i'll post solutions if you want

 

[hermine]

i'll post solutions if you want

ahh that would be a big help, thankyou

 

[lolokay]

aww you sure you wouldn't just prefer hints so you can solve them yourself?

solutions:
1.

Spoiler

there are 7! ways to arrange UNIFORM, and there is a 1/2 chance (ie. half of the arrangements) that M is right of U
so there is 7!/2 arrangements

2.

Spoiler

there are 5 ways for the brother/sister to choose the hostel
there are then 4 ways for each of the 3 backpackers to choose
so there are 5*4*4*4 total arrangements

3.

Spoiler

if there are 4 digits, there are 2 possible numbers (1 or 3) that can go in the first spot
there are then 4 for the second, 3 for the third, 2 for the fourth .'. 2*4*3*2 ways
if there are 3 digits you can have 5 ways to choose the first spot, 4 in the second, 3 in the third, so 5*4*3 ways
if there are 2 digits you can have 5 ways to choose the first spot, then 4 in second so 5*4 ways
if there is 1 digits you can have 5 possible digits in the first spot so 5 ways
add them up and you get 133

my solution to the third one is probably worded quite poorly

 

[hermine]

Alright, before i look at the solutions this is what i keep getting for each question

1) For the word UNIFORM, how many arrangements are possible if M is somewhere to the right of U?

I somehow figured out there was 6 ways for M to be right of U then x 6! to get the total arrangement

OH I just figured out I was figuring out ways for M to be directly, not somewhere right of U. ah fail haha


2) Five Backpackersarrive in a city with five youth hostels. Suppose two backpackers are brother and sister, so wish to stay in the same hostel. How many different accomodation arrangements are there if if the other three can go to any of the other hotels?

The brother and sister are in one hotel, and there are 4 hotels left for the three others
so i did 1 x 4^3


3) Numbers less than 4000 are formed from the digits 1, 3, 5, 8 and 9. How many numbers possible?

Only 1 or 3 can be the first number for it to be less than 4000 (2!)
and the remaining 3 digits are arranged in 4P3 ways
so 2 x 4P3


all of which =soo wrong lol maaaaan

 

[hermine]

solutions: ...[

ahhhh alright
1. How did you gat that 1/2 probability

2. ohhhhhhhhhh makes so much more sense

3. " " "

 

[lolokay]

well the U has to be either to the right or the left of M, and there would be the same number of arrangements of each -> so half are to the right, half to the left. (I meant to include that in the solution)

 

[hermine]

ah alright, thanks for that. such simple questions and I still didnt get it lol, you've saved my sanity for tonight