Question

is this a correct tree diagram

Answer 1

The Gladstone harbour board decides to issue private boats uusing the Gladstone harbour with 7 digit registration numbers. None of these can start with zero. The harbour board had only been able to get the digits 0, 1, 2, 3, 4, 5, 7, 9 for the boat registration numbers. No digit may be used more than once in each registration number.
Bruno, who was responsible for allocating the registration numbers to the boats, had the bright idea of using the number 9 upside down to make 6. However, the 6 and 9 cannot both occur in th same registration number.
The registration numbers are issued in ascending orider. Thus the first registration number is 1023567. The next two are 1023576 and 1023679.
How many boats can have registration numbers staring with 6.

Answer 2

looks good to me

Answer 3

so there are 5040 outcomes

Answer 4

does it make sense to you?

Answer 5

yes. cool. could u help me with another question. it just gets more difficult . :(

Answer 6

This looks right

Answer 7

fire away

Answer 8

How many boats can have registration numbers staring with 1?

Answer 9

good one... this will be slightly different from the first. you want to give it a shot?

Answer 10

i have got this so far

Answer 11

(7x6x5x4x3x2) - ?

Answer 12

I need to find a way to put 6 without 9 and vice versa

Answer 13

use the inclusion/exclusion principle

Answer 14

how about this...
\[\left(\begin{matrix}1 & \alpha & \beta & \gamma & \delta & \epsilon & \eta\\ 1st & 2nd & 3rd & 4th & 5th & 6th & 7th\end{matrix}\right)\]

if we exclude 6 and 9, the it will be 6!

then we have ot include the 6 and 9 options in all the different spots so we'll have to add to this

Answer 15

can u show me some of the working out i am confused

Answer 16

Do you get the 6! whwne we exclude 6 & 9 from our choices?

Answer 17

I dont know how to do it.

Answer 18

9 is also a possibility now

Answer 19

then you'll have 6 spots in which you can include 6 or 9, with 6P5 choices in the remaining 5 spots. so you'll add 6(2*6!) to the previous 6! for a total of 13*6!

Answer 20

6 x 12 = 72

Answer 21

case where a 6 or 9 is chosen + cases where a 6 or 9 isn't chosen

Answer 22

72?

Answer 23

so do i add 6(2*6!)

Answer 24

here is what i have done so far

Answer 25

yes, 12*6! sequences containing 6 or 9 + 6! sequences not including 6 or 9

Answer 26

no, not that picture

Answer 27

i thought there were 7 digits in the registration number...

Answer 28

oh yeah, i am confused would the last one have 6 and 9 on it

Answer 29

is this right?

Answer 30

no, look at this and see if it makes sense...

Answer 31

okay so i should just write up all these and add it together?

Answer 32

yeah, there's 6 spots for the 6 or 9 to go and the 6 & 9 make for 2 choices with 6! for the remaining choices, so that's where the 6*2*6! comes from... get it?

then when the 6 and 9 are out of the picture, there's just 6! arrangenments (the first table in the attachement).

does this make sense?

Answer 33

okay i will type it up

Answer 34

alright is this right?

Answer 35

yeah, you just have to sum all of those and you should get 13*6!

Answer 36

k

Answer 37

here is what i got

Answer 38

that's it

Answer 39

thanks!

Answer 40

you're welcome

Answer 41

is there a smaller formula i can use a second method i can show

Answer 42

there may be but I don't know what it is

Answer 43

k
i know this is asking a lot but can u show me how to find the totla number of possible registration numbers

Answer 44

yeah, it will be in a similar way as we just did the last one. it can't start with 0 right?

Answer 45

yes

Answer 46

okay and it can't have both a 6 and a 9, right?

Answer 47

yes

Answer 48

so do the first table in a similar way where 6 and 9 are excluded and then see how it will change when 6 or 9 is allowed

Answer 49

so should i write out all the tables :O

Answer 50

no, just the first and maybe a couple more to see the pattern

Answer 51

would it not be this: 9360 outcomes x 6 + 5040 outcomes

Answer 52

? where is that coming from?

Answer 53

9360 outcomes comes from the outcomes of starting with 1

Answer 54

yes that