# Simple Permutations and Combinations Problem

• Mar 17th 2014, 05:02 AM
zikcau25
Simple Permutations and Combinations Problem
I quote a simple question as follows but having trouble getting to the suggested answer, kinda boring:

Quote:

(i) Find the number of ways that a set of 10 different CDs can be shared between Dai and Evan if each receives an odd number of CDs.

ANS: 512

Need help on that, Thanks. (Nod)
• Mar 17th 2014, 06:23 AM
Plato
Re: Simple Permutations and Combinations Problem
Quote:

Originally Posted by zikcau25
I quote a simple question as follows but having trouble getting to the suggested answer,

Have a look at this.

You tell us why that works.
• Mar 17th 2014, 09:26 AM
zikcau25
Re: Simple Permutations and Combinations Problem
That what I would understand by " the 10 CDs Shared between Dai and Evan in odd number of CDs" as two separate and odd selections of CDs that always add up to 10 CDs (complementary), each for Dai and Evan.

I found out these possible5 ways of combining the two selections as follows:

1. $(1_{Dai}, 9_{Evan})$

$\binom{10}{1}\times \binom{9}{9}=10$

2. $(3_{Dai}, 7_{Evan})$

$\binom{10}{3}\times \binom{7}{7}=120$

3. $(5_{Dai}, 5_{Evan})$ = $(5_{Evan}, 5_{Dai})$

$\binom{10}{5}\times \binom{5}{5}=252$

4. $(7_{Evan}, 3_{Dai})$

$\binom{10}{7}\times \binom{3}{3}=120$

5. $(9_{Evan}, 1_{Dai})$

$\binom{10}{9}\times \binom{1}{1}=10$

Therefore, Sum of all Outcomes = $10+120+252+120+10 = 512$
• Mar 17th 2014, 10:01 AM
Plato
Re: Simple Permutations and Combinations Problem
Quote:

Originally Posted by zikcau25
That what I would understand by " the 10 CDs Shared between Dai and Evan in odd number of CDs" as two separate and odd selections of CDs that always add up to 10 CDs (complementary), each for Dai and Evan.

There is a more systematic way to do this.
If you have a set of ten there are $2^{10}$ subsets of the set. Half have an even number of elements the other have odd number.
If Dai gets an odd number of elements then there are an odd number of elements are left for Evan.

$\dfrac{2^{10}}{2}=2^9=512$
• Mar 17th 2014, 01:42 PM
Soroban
Re: Simple Permutations and Combinations Problem
Hello, zikcau25!

Quote:

Find the number of ways that a set of 10 different CDs
can be shared by Dai and Evan if each receives an odd number of CDs.

. . $\begin{array}{ccccc} \text{Dai} & \text{Evan} \\ \hline 1 & 9 & {10\choose1,9} &=& \;10 \\ 3&7 & {10\choose3,7} &=& 120 \\ 5&5 & {10\choose5,5} &=& 252 \\ 7&3 & {10\choose7,3} &=& 120 \\ 9&1 & {10\choose9,1} &=& \;10 \\ \hline &&&& 512\end{array}$
• Mar 17th 2014, 11:17 PM
zikcau25
Re: Simple Permutations and Combinations Problem
Thanks for the correction.
I think I was wrong in swapping the names in the other half, because the list is naturally symmetrical.
Now it looks clearer and direct.