Selection from categorized items

5

1. The menu for a meal in a restaurant is as follows. \begin{displayquote} Starter Course
   Melon
   or
   Soup
   or
   Smoked Salmon \end{displayquote} \begin{displayquote} Main Course
   Chicken
   or
   Steak
   or
   Lamb Cutlets
   or
   Vegetable Curry
   or
   Fish \end{displayquote} \begin{displayquote} Dessert Course
   Cheesecake
   or
   Ice Cream
   or
   Apple Pie
   All the main courses are served with salad and either
   new potatoes or french fries.
   1. How many different three-course meals are there?
   2. How many different choices are there if customers may choose only two of the three courses?
2. In how many ways can a group of 14 people eating at the restaurant be divided between three tables seating 5, 5 and 4? \end{displayquote}

7

1. A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done
   1. if the captain, who is a midfield player, must be included, together with one defence and one forward player,
   2. if exactly one forward player must be included, together with any two others.
2. Find how many different arrangements there are of the nine letters in the words GOLD MEDAL
   1. if there are no restrictions on the order of the letters,
   2. if the two letters D come first and the two letters L come last.

5 An English examination consists of 8 questions in Part \(A\) and 3 questions in Part \(B\). Candidates must choose 6 questions. The order in which questions are chosen does not matter. Find the number of ways in which the 6 questions can be chosen in each of the following cases.

1. There are no restrictions on which questions can be chosen.
2. Candidates must choose at least 4 questions from Part \(A\).
3. Candidates must either choose both question 1 and question 2 in Part \(A\), or choose neither of these questions.

4 An examination paper consists of three sections.

• Section A contains 6 questions of which the candidate must answer 3
• Section B contains 7 questions of which the candidate must answer 4
• Section C contains 8 questions of which the candidate must answer 5
  1. In how many ways can a candidate choose 3 questions from Section A?
  2. In how many ways can a candidate choose 3 questions from Section A, 4 from Section B and 5 from Section C?

A candidate does not read the instructions and selects 12 questions at random.

Find the probability that they happen to be 3 from Section A, 4 from Section B and 5 from Section C.

4 Peter and Esther visit a restaurant for a three-course meal. On the menu there are 4 starters, 5 main courses and 3 sweets. Peter and Esther each order a starter, a main course and a sweet.

1. Calculate the number of ways in which Peter may choose his three-course meal.
2. Suppose that Peter and Esther choose different dishes from each other.
   (A) Show that the number of possible combinations of starters is 6 .
   (B) Calculate the number of possible combinations of 6 dishes for both meals.
3. Suppose instead that Peter and Esther choose their dishes independently.
   (A) Write down the probability that they choose the same main course.
   (B) Find the probability that they choose different dishes from each other for every course.

2 An examination paper consists of two sections. Section A has 5 questions and Section B has 9 questions. Candidates are required to answer 6 questions.

1. In how many different ways can a candidate choose 6 questions, if 3 are from Section A and 3 are from Section B?
2. Another candidate randomly chooses 6 questions to answer. Find the probability that this candidate chooses 3 questions from each section.

The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics{figure_6} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.

1. How many selections of seven cards are possible? [3]
2. Find the probability that the seven cards include exactly one card showing the letter A. [4]