Question

Question #26b22

Answer 1

Use the complement method for an easy solution.

Explanation:

First, determine the number of ways that 4 and 5 ARE next to each other:

45xxxx
x45xxx
xx45xx
xxx45x
xxxx45

So, these 5 ways plus the 5 reverse ways where 5 precedes 4.

In total, there are 10 ways that the numbers 4 and 5 are next to each other. However, there are still 4! = 24 ways to arrange the remaining 4 numbers. So, there are `10 * 24 = 240` ways that 4 and 5 ARE next to each other .

In general, there are 6! = 720 ways to arrange the 6 numbers.

Using the complement method :

P(4 and 5 not next to each other) `= 1 - 240/720 = 2/3`

Hope that helped

Answer 2

Prob. that 4 and 5 are always next to each other is `2xx5!` / 6! = 1/3.
Probability that cards numbered 4 and 5 are not next to each other is 1 - 1/3 = 2/3.

Explanation:

The events in question are complements.