 # 1. Sailing ships used to send messages with signal flags flown from their masts. How many different Jazlyn Raymond 2022-05-08 Answered
1. Sailing ships used to send messages 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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For question 1, we must understand that with a minimum of 2 flags and a maximum of 4 flags being used for each signal configuration, the following configurations are possible:

1. 2 Flags Used
2. 3 Flags Used
3. 4 Flags Used

In the case of only 2 flags being used, any 4 of the flags can be picked initially as the possible first flag, leaving 3 flags remaining as the possible second flag. Thus, the number of different signals that can be made here is:
$4\cdot \left(4-1\right)=4\cdot 3=12$
In the case of using 3 flags, any 4 of the flags can be picked initially as the possible first flag, leaving 3 flags remaining as the possible second flag, leaving 2 flags remaining as the possible third flag. Thus, the number of different signals that can be made here is:
$4\cdot \left(4-1\right)\cdot \left(4-2\right)=4\cdot 3\cdot 2=24$
In the final case of using all 4 flags, any 4 of the flags can be picked initially as the possible first flag, leaving 3 flags remaining as the possible second flag, leaving 2 flags remaining as the possible third flag, leaving only 1 flag as the final flag to be chosen. Thus the number of different signals that can be made here is:
$4\cdot \left(4-1\right)\cdot \left(4-2\right)\cdot \left(4-3\right)=4\cdot 3\cdot 2\cdot 1=24$
$12+24+24=60$
For question 2, we can think of a tree-diagram to represent this. Starting with 3 separate trees: Science, Geography, and Phys. Ed at the top. From here, there are 4 possible classes to branch off of the first class: Art, Music, French, or business. We then branch off of these classes to Math or English, and the final period will be Math if the third period was English, or English if the third period was Math, in order to ensure no duplicates.

From here, $3\cdot 4\cdot 2=24$ possible timetables can be chosen.