Counting Principles

I am struggling with additive counting principle with following questions

1. Sailing ships used to send massages with signal flags flown from their masts. How many different signals are possible with a set of four distinct flags if a minimum of two flags is used for each signals?

2. A Gr. 9 students may build a timetable by selecting one course for each period, with no duplication of courses. Period 1 must be science, geography, or physical education. Period 2 must be art, music, French, os business. Period 3 and 4 must be math or English. How many different timetables could a student choose?

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F>question number one. We will count the messages that use 2 flags, 3flags and 4 flags seperately.

When choosing 2 flags out of four you first have 4 options and then 3. however you need to divide by 2 since the order in which you picked them is irrelevant. for example we are counting the case in which you pick the red flag first and then the blue and the case in which you picked the blue one and then the red as two different cases, when they are the same one. choosing three out of four is the same as choosing which one you don't want, therefore there are 4 messages with 3 flags. and the message with 4 flags is unique since you use all 4 of 4. So then there are 6 messages with 2 flags.

adding all these messages we see there are 11 total.

question 2

for period one you have 3 choices. For period two you have 4. in period three you have 2 choices and in period four you have to take the class you didn't take in period three.

Therefore the total number is $3*4*2=24$

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in question 2 for period 2 students have choice of 4 subject, why you count it three? – Sunita Mar 20 '13 at 11:21
sorry, I got confused, it is four. It looks like you get it though – Carry on Smiling Mar 20 '13 at 12:23