# Thread: Permutations and Combinations Marathon

1. ## Permutations and Combinations Marathon

To start off:
The ratio of numbers of arrangements of (2n+2) different objects taken n at a time to the number of arrangements of 2n different objects n at a time is 14:5. Find the value of n.

2. ## Re: Permutations and Combinations Marathon

Originally Posted by Rathin
To start off:
The ratio of numbers of arrangements of (2n+2) different objects taken n at a time to the number of arrangements of 2n different objects n at a time is 14:5. Find the value of n.
(2n+2)Cn / 2nCn = 14/5
(2n+2)!/n!(2n+2-n)! * n!(2n-n)!/(2n)! = 14/5
(2n+2)!/n!(n+2)! * (n!)^2 / (2n)! = 14/5
(2n+2)(2n+1)/(n+2)(n+1) = 14/5
5(2n+2)(2n+1) = 14(n+2)(n+1)
5(4n^2 + 6n + 2) = 14(n^2 + 3n + 2)
20n^2 + 30n + 10 = 14n^2 + 42n + 28
6n^2 - 12n - 18 = 0
n^2 - 2n - 3 = 0
(n-3)(n+1) = 0
n = -1, 3 but n>0
n = 3

EDIT: It's meant to be perms not combs but still gives the same answer as dividing those 2 combinations gets rid of the n! which is the difference between the P and C.

3. ## Re: Permutations and Combinations Marathon

In how many ways can 5 writers and 5 artists be arranged in a circle so that two particular artists must not sit next to a particular writer. Note: The writers and artists sit alternately.

4. ## Re: Permutations and Combinations Marathon

Originally Posted by Rathin
In how many ways can 5 writers and 5 artists be arranged in a circle so that two particular artists must not sit next to a particular writer. Note: The writers and artists sit alternately.
4!*5! = 2880

5. ## Re: Permutations and Combinations Marathon

Originally Posted by bujolover
4!*5! = 2880
That is for when the artists are seated alternatively only. However there are other conditions to the question.

6. ## Re: Permutations and Combinations Marathon

Fix the writer = 1
Sit 2 artists next to him = 3 x 2
Sit the rest of the writers = 4!
Sit the rest of the artists = 3!

Multiply to get 864

7. ## Re: Permutations and Combinations Marathon

Originally Posted by pikachu975
Fix the writer = 1
Sit 2 artists next to him = 3 x 2
Sit the rest of the writers = 4!
Sit the rest of the artists = 3!

Multiply to get 864
Yup that is correct!

8. ## Re: Permutations and Combinations Marathon

Originally Posted by kawaiipotato
$\noindent A pack of sweets contains 25 sweets, a combination of mini-cupcakes and mini-macarons. If there are 18 varieties of mini-cupcakes, and 10 varieties of mini-macarons, how many ways can you fill a pack? (The order does not matter, just how many of each sweet type you have in the pack.)$
From the HSC 2017 MX2 Marathon thread.

(I don't have the answer unfortunately.)

9. ## Re: Permutations and Combinations Marathon

Originally Posted by kawaiipotato
From the HSC 2017 MX2 Marathon thread.

(I don't have the answer unfortunately.)
I dunno if its this simple tbh but:
28C25 = 3276 ways

10. ## Re: Permutations and Combinations Marathon

How many ways can eight basketball players be divided into four groups of two?

11. ## Re: Permutations and Combinations Marathon

Originally Posted by Rathin
How many ways can eight basketball players be divided into four groups of two?
8C2 x 6C2 x 4C2 x 2C2 / 4! = 28x15x6 / 4! = 105

Divided by 4! because if you pick the same teams but arrange them in different order then it will give 4! arrangements, so you gotta get rid of those as if it's a repetition when arranging EEEE etc.

12. ## Re: Permutations and Combinations Marathon

A 3-player game is played between Andy, Ben and Chuck.
The probabilities that each of those players win a game are 0.5, 0.3 and 0.2 respectively.
There are no drawn games.
The winner of a tournament is the first player to win 6 games.
The current score in the tournament (ie. games won) is:
Andy - 3
Ben - 4
Chuck - 2
To 5 decimal places, what is the probability that Andy wins the tournament from here?

13. ## Re: Permutations and Combinations Marathon

Originally Posted by braintic
A 3-player game is played between Andy, Ben and Chuck.
The probabilities that each of those players win a game are 0.5, 0.3 and 0.2 respectively.
There are no drawn games.
The winner of a tournament is the first player to win 6 games.
The current score in the tournament (ie. games won) is:
Andy - 3
Ben - 4
Chuck - 2
To 5 decimal places, what is the probability that Andy wins the tournament from here?
Most likely completely wrong but is the answer 0.18075

14. ## Re: Permutations and Combinations Marathon

Originally Posted by pikachu975
Most likely completely wrong but is the answer 0.18075
Sorry, I forgot I asked this question, and I can't find my solution. I just tried it again and got 0.31275.

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