# Question 26b22

##### 2 Answers
Oct 14, 2015

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 $10 \cdot 24 = 240$ 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) $= 1 - \frac{240}{720} = \frac{2}{3}$

Hope that helped

Oct 15, 2015

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.