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.
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
Hello, CountNumberla!
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
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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