Hi I just wrote a probability midterm and would like to verify a couple of the short counting questions worth a point each.
I understand it's better to ask one question at a time but it seems like a waste to separate such short simple questions
Q1: 6 People are waiting for the bus, considering 2 of the people refuse to stand behind each other what is the total number of ways they can line up?
6! ways they can line up in general. If A lines up in first place that removes 4! combinations If A lines up in last place that removes 4! combinations If A lines up in place 2,3,4,5 that removes 2*4! combinations 4 times
I got 6! - 10x4! total
Q2: 3 balls are randomly dropped into 3 bins. What is the probability of exactly one bin being empty?
$3^3$ total combinations = 81
012 (3x2 combinations) = 6
021 (3x2 combinations) = 6
102 (3x2 combinations) = 6
201 (3x2 combinations) = 6
120 (3x2 combinations) = 6
210 (3x2 combinations) = 6
so $\frac{36}{81} = \frac{4}{9}$
Any mistakes?
For the second, $3^3=27,$ not $81$. When you count the number of combinations for $012$ you just need to choose the ball in the $1$, so there are only $3$ combinations. This gives $\frac {18}{27}=\frac 23$. As a check $111$ gives $6$ combinations and $003$ in some order gives $3$, making the total $27$.