Question

Counting 1. Our student clubs are being particularly active. The aquatics club in particular is in training for an upcoming competition. For all parts of this question, working is required, including combinatoric/factorial notation as needed as well as final answers. a) 12 12 5 marks] During training, there are several different types of laps that can be swum: freestyle arms only freestyle one arm only, left then right torpedo/kickboard (straight kicking legs only) backstroke breaststroke Assume that each type is undertaken at most once in a training session, over contiguous laps (i.e. so once you have completed your freestyle laps, you do not repeat freestyle). The number of laps for each lap type is unimportant, that is, 4 laps of freestyle followed by 2 laps of backstroke is the same as 3 laps of freestyle followed by 4 laps of backstroke. i. How many different lap type patterns can be formed from those six types? ii. How many patterns contain only three types (e.g., freestyle arms only- one arm only)? ili. How many patterns with at least four different types start with torpedo/kickboard laps? b)12+1 - 3 marks] The club is starting to look at putting together some relay teams. There are 15 swimmers nominating for this opportunity i. If there are three different races, and a team of 5 needs to be nominated for each race, how many different teams can be formed? ii. Show a second approach to your answer to i. c)12 marks] One of the club swimmers is putting their training plans together for the compe- tition that is 30 days from now. This swimmer trains no more than once a day. If they have 18 training sessions planned, explain using the counting principles covered in class how this means they will train on consecutive days at least once ahead of the competition

Answer 1

Answer

a)

Assumptions

Each type of lap is undertaken at most once, over contiguous lap.

The number of laps for each lap type is not important.

i)

There are total 6 types of laps.

As it is given that each lap is undertaken at most once, the patterns can be formed by taking 0/1/2/3/4/5/6 laps from the given 6 types of laps.

Pattern formed by taking no lap from the given 6 laps

It is equivalent to permutation of 6 items taken 0 at a time.

Pattern formed by taking 1 lap from the given 6 laps

It is equivalent to permutation of 6 items taken 1 at a time.

Pattern formed by taking 2 laps from the given 6 laps

It is equivalent to permutation of 6 items taken 2 at a time.

Pattern formed by taking 3 laps from the given 6 laps

It is equivalent to permutation of 6 items taken 3 at a time.

Pattern formed by taking 4 laps from the given 6 laps

It is equivalent to permutation of 6 items taken 4 at a time.

Pattern formed by taking 5 laps from the given 6 laps

It is equivalent to permutation of 6 items taken 5 at a time.

Pattern formed by taking 6 laps from the given 6 laps

It is equivalent to permutation of 6 items taken 6 at a time.

Total number of patterns(excluding the empty pattern) which can be formed from the given 6 types of laps is

ii)

Pattern formed by taking 3 laps from the given 6 laps

It is equivalent to permutation of 6 items taken 3 at a time.

iii)

Pattern formed by taking 4 laps from the given 6 laps starting with torpedo/kickboard

Here, the patterns always start with a specific type of lap. Hence, we can only permute the other 3 laps from 5 laps.

It is equivalent to permutation of 5 items taken 3 at a time.

Pattern formed by taking 5 laps from the given 6 laps starting with torpedo/kickboard

Here, the patterns always start with a specific type of lap. Hence, we can only permute the other 4 laps from 5 laps.

It is equivalent to permutation of 5 items taken 4 at a time.

Pattern formed by taking 6 laps from the given 6 laps starting with torpedo/kickboard

Here, the patterns always start with a specific type of lap. Hence, we can only permute the other 5 laps from 5 laps.

It is equivalent to permutation of 5 items taken 5 at a time.

Total number of patterns with at least 4 different types staring with torpedo/kickboard is

b)

i)

There are total 15 swimmers.

A team of 5 is needed for 3 races. Hence, total 3 teams are needed.

Total number of different teams that can be formed is equivalent to selecting 5 swimmers from 15 swimmers for the first team, then the next 5 swimmers from remaining 10 swimmers for the second team and at last the 5 swimmers from remaining 5 swimmers for the third team.

P.S. - Answered 4 sub-parts as HOMEWORKLIB guidelines.