1. After a year of not doing much, everyone has decided to get active. We thought about this in assignment 1 as well. For all parts of this question, working is required, including combinatoric/factorial notation as needed as well as final answers. An answer consisting of solely an integer will be awarded 0 marks.

a) ? [2 + 1 + 2 = 5 marks] During a given day, there are many activities that could be done, such as:

go shopping

get an immunisation

visit the gym

go to a tutorial

join a virtual student event.

Assume that each activity is undertaken at most once a day (i.e., so once you have visited the gym

you don't visit it again). The amount of time spent on each activity is also unimportant, however, for

example, shopping before a tutorial and a tutorial before shopping should be considered different.

We are interested in working out the different ways people spend their day.

i. How many different activity patterns can be formed from those five activities?

ii. How many patterns contain only three activities (e.g., shop - immunisation - gym)?

iii. How many patterns with at least four different activities start with going shopping?

b) ? [2 + 2 + 2 + 1 = 7 marks] You have twelve friends that you haven't seen in a year that you

would like to meet up with. For parts i-iii, your reasoning needs to be explained in a sentence

or two; 0 marks will be awarded for a number or expression alone.

i. You have won a voucher for 5 people to go indoor rockclimbing. How many options are there

for inviting your friends to join you?

ii. All twelve friends are invited to meet you at an outdoor gym for some exercise at a particular

time. How many outcomes are there from this invite?

iii. With eleven friends, you are going to a concert in the city but are required to sit in pods of four

at the venue. How many different ways can the pods be organised?

iv. Show a second approach to your answer to iii.

c) [3 marks] After all these social activities, you decide you probably should do some cleaning

at home, and put together a schedule for the next month (30 days). You don't clean more than

once a day. If you have 20 cleaning sessions planned, explain using the counting principles

covered in class how this means you will clean on consecutive days at least once in the next

month