Specific items separated

4

1. Find the number of different ways that the 9 letters of the word HAPPINESS can be arranged in a line.
2. The 9 letters of the word HAPPINESS are arranged in random order in a line. Find the probability that the 3 vowels (A, E, I) are not all next to each other.
3. Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no Ps and either one or two Ss.

3 Six red counters and four blue counters are arranged in a straight line in a random order.
Find the probability that

1. no blue counter has fewer than two red counters between it and the nearest other blue counter,
2. no two blue counters are next to one another.

1. a) 5 girls and 3 boys are arranged at random in a straight line. Find the probability that none of the boys is standing next to another boy.
   (3 marks)
   b) A cricket team consisting of six batsmen, four bowlers, and one wicket-keeper is to be selected from a group of 18 cricketers comprising nine batsmen, seven bowlers, and two wicket-keepers.
   How many different teams can be selected?
   (3 marks)
   [0pt] [BLANK PAGE]
2. \(\quad \mathrm { P } ( E ) = 0.25 , \mathrm { P } ( F ) = 0.4\) and \(\mathrm { P } ( E \cap F ) = 0.12\)
   a Find \(P \left( E ^ { \prime } \mid F ^ { \prime } \right)\)
   b Explain, showing your working, whether or not \(E\) and \(F\) are statistically independent. Give reasons for your answer.

The event \(G\) has \(\mathrm { P } ( G ) = 0.15\)
The events \(E\) and \(G\) are mutually exclusive and the events \(F\) and \(G\) are independent.
c Draw a Venn diagram to illustrate the events \(E , F\) and \(G\), giving the probabilities for each region.
d Find \(\mathrm { P } \left( [ F \cup G ] ^ { \prime } \right)\)
[0pt] [BLANK PAGE]