Possible Arrangements: A family has three children....

[shay]

A family has three children. I am to write down all the possible arrangements for three children oldest to youngest.

Do I have all of the possible arrangements B= boy G= girl; in order from oldest to youngest

bgb
bbg
gbb
gbg
ggb
bgg
bbb
ggg

 

[pka]

Yes you do. But it is better to be systemic about it.
BBB
BBG
BGB
BGG
GBB
GBG
GGB
GGG.

Look up how one makes a ‘TRUTH TABLE’.
Then use B & G instead of T & F.
You can also use this for coin tossing using T & H.

 

[Denis]

Re: Possible Arrangements

A family has three children. I am to write down all the possible arrangements for three children oldest to youngest.

What's "oldest to youngest" got to do with this?

 

[stapel]

Re: Possible Arrangements

What's "oldest to youngest" got to do with this?

If there are two girls and one boy, there are three different arrangement possible, depending upon whether the boy is the eldest, the middle, or the youngest child.

Eliz.

 

[soroban]

Hello, shay!

pka is absolutely correct . . . you need a system.

There are two possible "values": \(\displaystyle \,\{B,G\}\)

If there are \(\displaystyle n\) children, there are \(\displaystyle 2^n\) possible outcomes.


So with 3 children, there are \(\displaystyle 2^3\,=\,8\) outcomes.

We will make a 3-column table.

In column-1, write B's and G's four-at-a-time: \(\displaystyle \;\begin{array}{cccccccc}B\\B\\B\\B\\G\\G\\G\\G\end{array}\)

In column-2, write B's and G's two-at-a-time: \(\displaystyle \;\begin{array}{cccccccc}B\\B\\G\\G\\B\\B\\G\\G\end{array}\)

In column-3, write B's and G's one-at-a-time: \(\displaystyle \;\begin{array}{cccccccc}B\\G\\B\\G\\B\\G\\B\\G\end{array}\)


Reading across the table, we have all eight possible outcomes:

\(\displaystyle \;\;\;\begin{array}{cccccccc}1&B&B&B\\2&B&B&G\\3&B&G&B\\4&B&G&G\\5&G&B&B\\6&G&B&G\\7&G&G&B\\8&G&G&G\end{array}\)