Question

Probability that a comes before b and b comes before c

Answer 1

I worked out each part, but I am not really sure about my answers after part a.

How many different linear arrangements are there of the letters A B C D E F for which:

a. A and B are next to each other? 2!5! = 240

I grouped A and B as "one element" so now there's 5 elements, so 5! ways to arrange them, multiplied by the 2! ways to arrange AB (either AB or BA)

b. A is before B?

correct answer: 6!/2

1st attempt:

5!+4!+3!+2!+1!= 153

The way I pictured this problem is:

A _ _ _ _ _ 5! ways for B to be arranged

X A _ _ _ _ 4! ways for B to be arranged

X X A _ _ _ 3! ways for B to be arranged

X X X A _ _ 2! ways for B to be arranged

X X X X A _ 1! ways for B to be arranged

c. A is before B and B is before C?

1st attempt:

5!4!(3!)22!1!5!4!(3!)22!1!

5! ways to choose letter A

4! ways to choose letter B

3! ways to choose letter C

3! ways to choose letter 4

2! ways to choose letter 5

1! way to choose letter 6

d. A is before B and C is before D?

1st attempt:

(I am not at all confident about this answer)

5!4! ways for A to be before B, then 3!2! ways for C to be before D, multiplied by 2! because we could do the reverse (5!4! ways for C to be before D, then 3!2! ways for A to be before B

e. A and B are next to each other and C and D are also next to each other? = 4!2!2!4!2!2!

So I am grouping AB as "one element" and CD as one element", so there are now 4 element.

4! ways to arrange them

2! ways to arrange AB

2! ways to arrange CD

f. E is not the last in line? = (5!)24!3!2!1!(5!)24!3!2!1!

5! ways to choose E

5! ways to choose letter 2

4! ways to choose letter 3

3! ways to choose letter 4

2! ways to choose letter 5

1! ways to choose letter 6