# Thread: Mixed bag part 2: pemutations, combinations and determinants

1. ## Mixed bag part 2: pemutations, combinations and determinants

I just want to make sure I have my method down properly, as usual, so here goes another round of questions.

1:
There are 10 women and 8 men in a club. How many different committees of 6 people can be selected from the group if equal numbers of men and women are to be on the committee?

Method:

There are$\displaystyle _{10}W_3$ ways to select 3 women for the committee and$\displaystyle _8M_3$ ways to select 3 men for the committee. Therefore, there are $\displaystyle _{10}W_3\cdot_8M_3$different committees possible, so $\displaystyle _{10}W_3 \cdot _8M_3 \rightarrow \frac{10!}{(10-3)!} \cdot \frac{8!}{(8-3)!}\rightarrow\frac{10!}{7!}\cdot\frac{8!}{5!}\ri ghtarrow$$\displaystyle \frac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3 \cdot2\cdot1}{7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot 1}\cdot\frac{8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2 \cdot1}{5\cdot4\cdot3\cdot2\cdot1}\rightarrow\frac {3628800}{5040}\cdot\frac{40320}{120}=241920$
Thus there are 241920 possible committees.

2:
Two marbles are drawn from a bag containing 6 white, 4 red, and 6 green marbles. Find the probability of both marbles being white.

Method:
There are 16 marbles in all so there are $\displaystyle _{16}M_2\rightarrow\frac{16!}{14!}\rightarrow \frac{2.09227899 \cdot 10^{13}}{87178291200}=240$ways of picking 2 marbles.
This is where I get fuzzy, I'm not too sure what to do next--should I multiply out all the permutations for all the colors and then work from there?
Or should I just focus on the white marbles?

3:
If two rows of a determinant are interchanged, what is true of the resulting determinant?

Method:
Sadly for me, I have little clue of what to do. I could tell you why if one row of a determinant is 0 the whole things value is 0, but I don't know where to start on this one.

Thank you

2. Hello, MathBlaster47!

The first two are Combination problems, not Permutations.

1) There are 10 women and 8 men in a club.
How many different committees of 6 people can be selected from the group
if 3 women and 3 women are to be on the committee?
There are: .$\displaystyle _{10}C_3 \:=\:{10\choose3} \:=\:\frac{10!}{3!\,7!} \:=\:120$ ways to choose 3 women.

There are: .$\displaystyle _8C_3 \:=\:{8\choose3} \:=\:\frac{8!}{3!\,5!} \:=\:56$ ways to choose 3 men.

Therefore, there are: .$\displaystyle 120\cdot56 \:=\:6720$ committees with 3 women and 3 men.

2) Two marbles are drawn from a bag containing 6 white, 4 red, and 6 green marbles.
Find the probability of both marbles being white.

There are: .$\displaystyle _{16}C_2 \:=\:{16\choose2} \:=\:\frac{16!}{2!\,14!} \:=\:120$ possible two-marbles samples.

There are: .$\displaystyle _6C_2 \:=\:{6\choose2} \:=\:\frac{6!}{2!\,4!} \:=\: 15$ ways to get two white marbles.

Therefore: .$\displaystyle P(\text{two whites}) \;=\;\frac{15}{120} \;=\;\frac{1}{8}$

3) If two rows of a determinant are interchanged,
what is true of the resulting determinant?
It depends on which two rows are interchanged.

If the two rows are an odd number of rows apart,
. . the sign of the determinant is changed.

If the two rows are an even number of rows apart,
. . the value of the determinant is unchanged.

3. Wow! On exam 11 and 12! Almost there~! Chapter 16 was a real mo' fo' for me for whatever reason.. Anywho, thanks for asking these questions, I was a little unsure too, but my answers were the same as Soroban's *insert nerdy math gloating here* (rofl)

But for the last question;
If two rows of a determinant are interchanged, what is true of the resulting determinant?

I wouldn't put the "If the two rows are an even number of rows apart, the value of the determinant is unchanged." part Soroban's answer (no offense), because our textbook simply doesn't cover it. That chapter really only discussed 3 x 3 determinants. Look at chapter 16-8 Properties of Determinants (page 798) Property 3
"If two rows (or two columns) of a determinant are interchanged, then the resulting determinant is the opposite of the original determinant."

Even though that's not true for all determinants as Soroban pointed out, that's how they want you to answer D':

4. Originally Posted by StonerPenguin
Wow! On exam 11 and 12! Almost there~! Chapter 16 was a real mo' fo' for me for whatever reason.. Anywho, thanks for asking these questions, I was a little unsure too, but my answers were the same as Soroban's *insert nerdy math gloating here* (rofl)

But for the last question;
If two rows of a determinant are interchanged, what is true of the resulting determinant?

I wouldn't put the "If the two rows are an even number of rows apart, the value of the determinant is unchanged." part Soroban's answer (no offense), because our textbook simply doesn't cover it. That chapter really only discussed 3 x 3 determinants. Look at chapter 16-8 Properties of Determinants (page 798) Property 3
"If two rows (or two columns) of a determinant are interchanged, then the resulting determinant is the opposite of the original determinant."

Even though that's not true for all determinants as Soroban pointed out, that's how they want you to answer D':
Ah, thank you! I don't mind gloating--you did earn it!
I however, did not....alas those permutations and combination problems are not my strong suit, statistics, and much of trig I can do no problem.
Thanks for catching my oversight!

Originally Posted by Soroban
Hello, MathBlaster47!

The first two are Combination problems, not Permutations.

There are: .$\displaystyle _{10}C_3 \:=\:{10\choose3} \:=\:\frac{10!}{3!\,7!} \:=\:120$ ways to choose 3 women.

There are: .$\displaystyle _8C_3 \:=\:{8\choose3} \:=\:\frac{8!}{3!\,5!} \:=\:56$ ways to choose 3 men.

Therefore, there are: .$\displaystyle 120\cdot56 \:=\:6720$ committees with 3 women and 3 men.

There are: .$\displaystyle _{16}C_2 \:=\:{16\choose2} \:=\:\frac{16!}{2!\,14!} \:=\:120$ possible two-marbles samples.

There are: .$\displaystyle _6C_2 \:=\:{6\choose2} \:=\:\frac{6!}{2!\,4!} \:=\: 15$ ways to get two white marbles.

Therefore: .$\displaystyle P(\text{two whites}) \;=\;\frac{15}{120} \;=\;\frac{1}{8}$

It depends on which two rows are interchanged.

If the two rows are an odd number of rows apart,
. . the sign of the determinant is changed.

If the two rows are an even number of rows apart,
. . the value of the determinant is unchanged.

Thank you!
I realize where I went wrong....dang things looking so similar.....

5. Eh, that's cool. Determinants and matrices really aren't my suit so I know how it is. And the important thing is that you realize your mistakes, which is the key to improvement. And the users on here are really good at showing their process and helping, which is why I this forum lawl