Permutation and Combinations Help! (1 Viewer)

[Kujah]

Got some questions that I need assistance with.


1. The ratio of the number of combinations of (2n + 2) different objects taken "n" at a time to the number of combintations of (2n - 2) different objects taken "n" at a time is 99:7. Find the value of "n".


2. How many arrangements of the letters in tomato are there, if the letters o are to be separated.


3. 4 men, 2 women and 1 child sit at a round table. In how many ways can these 7 people be arranged

i. if the child is seated between two women
ii. if the child is seated between two men


4. In how many ways can 5 men and 5 women be arranged in a circle so that two particular women must not be next to a particular man?

 

[Shadowulf]

answering Q2.

Tomato has 6 letters, with 2 t's and 2 o's, so youll start off with

6! / 2!2! = 180
thats for every combination, now you simply want to minus every possibility of the Os being together. To do this, you'll group the 2 o's as 1 unit.

Then you'll have:
5! / 2! = 60

180-60 = 120 solutions

Q3

i. You'll have to count the child and 2 women as 1 group/person.
so you will be counting 5 groups sitting at a round table:

5!2! (in case youre wondering why there is an extra 2!, its because theres a woman on either side of the child, so there are two possible arrangements around the child)

= 240

ii. again, you will be grouping the child and 2 men as one, but since there are 4 men, you will have

5!(4x3)= 1440
, you have 4x3 because, for example, you will have 1 of 4 men sitting on one side of the child, and another of the remaining 3 men on the other side.

thats all i can answer

also, im not sure whether i even got question 3 right, i havent done this topic for more than a year, so im a bit rusty.
So if anyone can tell me whether im wrong or not, and give a proper explaination, come help me too lol

Thanks for the practise =)

EDIT:

Im gonna have a go at question 4

since there are 5 men and 5 women, first taking the total possibilities: 9!

Now assuming that this means that the 2 particular women do not sit next to the particular man at the same time; posible arrangements for the 2 women to sit next to that 1 man is:
7!2!

9! - 7!2! = 352800
looks like an insane number =S

Someone who actually remembers how to do this stuff plz teach us haha

 

[pLuvia]

1. Taking (2n+2) from n is nC(2n+2) and (2n-2) from n is nC(2n-2) and the ratio is 99:7 then,
nC(2n+2)/nC(2n-2)=99/7

Cancel out some of the common factors

[n!/(2n+2)!(-n-2)!]/[n!/(2n-2)!(2-n)!]=99/7

Can you work it out from there?

 

[Kujah]

1. Taking (2n+2) from n is nC(2n+2) and (2n-2) from n is nC(2n-2) and the ratio is 99:7 then,
nC(2n+2)/nC(2n-2)=99/7

Cancel out some of the common factors

[n!/(2n+2)!(-n-2)!]/[n!/(2n-2)!(2-n)!]=99/7

Can you work it out from there?

I got that.

Im gonna have a go at question 4

since there are 5 men and 5 women, first taking the total possibilities: 9!

Now assuming that this means that the 2 particular women do not sit next to the particular man at the same time; posible arrangements for the 2 women to sit next to that 1 man is:
7!2!

9! - 7!2! = 352800
looks like an insane number =S

Oops, I forgot to give you the full question.

In how many ways can 5 men and 5 women be arranged in a circle so that the men are separated? In how many ways can thise be done if two particular women must not be next to a particular man?


Thanks for our answers =)

 

[Kujah]

Got it!

3P2 (as the two women can't sit next to the fixed man, leaving 3 women to choose from to select 2 women) x 4! (arrangements of men) x 3! (arrangements of women)
= 864


Thanks guys, you've been great help !