Selection with type constraints

3

1. In how many ways can all 9 letters of the word TELEPHONE be arranged in a line if the letters P and L must be at the ends? How many different selections of 4 letters can be made from the 9 letters of the word TELEPHONE if
2. there are no Es,
3. there is exactly 1 E ,
4. there are no restrictions?

1. There are 16 competitors in a table-tennis competition, 5 of which come from Racknor Comprehensive School. Prizes are awarded to the competitors finishing in each of first, second and third place.

Assuming that all the competitors have an equal chance of success, find the probability that the students from Racknor Comprehensive

1. win no prizes,
2. win the \(1 ^ { \text {st } }\) and \(3 ^ { \text {rd } }\) place prizes but not the \(2 ^ { \text {nd } }\) place prize,
3. win exactly one of the prizes.

The 50 members of a club include both the club president and the club treasurer. All 50 members want to go on a coach tour, but the coach only has room for 45 people. In how many ways can 45 members be chosen if both the club president and the club treasurer must be included? [3]

1. State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey. [2]
2. State one advantage and one disadvantage of quota sampling. [2]

(Total 4 marks)

A washing-up bowl contains 6 spoons, 5 forks and 3 knives. Three of these 14 items are removed at random, without replacement. Find the probability that

1. all three items are of different kinds, [3]
2. all three items are of the same kind. [3]

There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.

1. In how many different ways can the three men be selected? [2]
2. In how many different ways can the whole committee be selected? [2]
3. A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women. [2]