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)

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.

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 possible**5** ways of combining the two selections as follows:

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

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

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

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

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

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

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

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

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

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

Therefore, Sum of all Outcomes = $\displaystyle 10+120+252+120+10 = 512$

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.

$\dfrac{2^{10}}{2}=2^9=512$

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.

. . $\displaystyle \begin{array}{ccccc} \text{Dai} & \text{Evan} \\ \hline 1 & 9 & {10\choose1,9} &=& \;10 \\ 3&7 & {10\choose3,7} &=& 120 \\ 5&5 & {10\choose5,5} &=& 252 \\ 7&3 & {10\choose7,3} &=& 120 \\ 9&1 & {10\choose9,1} &=& \;10 \\ \hline &&&& 512\end{array}$

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.