# [Probability] Seating arrangement & Selection

• Nov 20th 2009, 07:36 PM
CountNumberla
[Probability] Seating arrangement & Selection
PROBLEM 1
7 people sit in a row, and 2 of them want to sit together, in how many ways can they be arranged?

ATTEMPTED SOLUTION:
6! x 2 = 1440
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PROBLEM 2
7 people sit in a row, and 2 of them do not want to sit together, in how many ways can they be arranged?

ATTEMPTED SOLUTION:
No ideas
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PROBLEM 3
7 people sit in a circular formation, but 2 of them do NOT want to sit together, in how many ways can they be arranged?

ATTEMPTED SOLUTION
No ideas
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PROBLEM 4
There are 10 red & 6 blue marbles in a bag. You pick 2 of them at the same time.
In how many ways can you pick 2 marbles of the same color? Different colors?

ATTEMPTED SOLUTION:
Same colors: 10/16 x 9/15 + 6/16 x 5/15
Different colors: 10/16 x 6/16 + 9/15 x 5/15
• Nov 21st 2009, 03:20 AM
galactus
Your #1 is correct. For #2, subtract #1 from the total number of arrangements,which is 7!.

For the circular. There are (n-1)! ways to arrange n people around a circular table. Use the same ideas you had on the first problem where they're in a line.
• Nov 21st 2009, 04:07 AM
Soroban
Hello, CountNumberla!

Quote:

PROBLEM 4
There are 10 red & 6 blue marbles in a bag.
You pick 2 of them at the same time.
In how many ways can you pick 2 marbles of the same color?
Different colors?

ATTEMPTED SOLUTION:

Same colors: 10/16 x 9/15 + 6/16 x 5/15 . . . . Right!

Different colors: 10/16 x 6/16 + 9/15 x 5/15 . . . . no

$\displaystyle \text{Same colors: }\;\left(\frac{10}{16}\cdot\frac{9}{15}\right) + \left(\frac{6}{16}\cdot\frac{5}{15}\right) \quad=\quad \frac{1}{2}$
. . . . . . . . . . .$\displaystyle {\color{red}RR} \quad\;\; \text{or} \qquad {\color{blue}BB}$

$\displaystyle \text{Different colors: }\;\left(\frac{10}{16}\cdot\frac{6}{15}\right) + \left(\frac{6}{16}\cdot\frac{10}{15}\right) \quad=\quad \frac{1}{2}$
. . . . . . . . . . . . .$\displaystyle {\color{red}R}{\color{blue}B} \quad\;\;\text{ or } \quad\;\; {\color{blue}B}{\color{red}R}$

• Nov 21st 2009, 05:01 PM
CountNumberla
Thanks everybody!
Ok, here are my updated solutions:

PROBLEM 1
7 people sit in a row, and 2 of them want to sit together, in how many ways can they be arranged?

ATTEMPTED SOLUTION:
6! x 2 = 1440
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PROBLEM 2
7 people sit in a row, and 2 of them do not want to sit together, in how many ways can they be arranged?

ATTEMPTED SOLUTION:
7! - (6! x 2) = 3600
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PROBLEM 3
7 people sit in a circular formation, but 2 of them do NOT want to sit together, in how many ways can they be arranged?

ATTEMPTED SOLUTION
7! - (5! x 2) = 4800
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PROBLEM 4
There are 10 red & 6 blue marbles in a bag. You pick 2 of them at the same time.
In how many ways can you pick 2 marbles of the same color? Different colors?

ATTEMPTED SOLUTION:
Same colors: 10/16 x 9/15 + 6/16 x 5/15 = 1/2
Different colors: 10/16 x 6/15 + 6/16 x 10/15 = 1/2