probability.

[tecklee89]

A mother prepares nine popsicles of different flavours:
3 of orange,
3 of cherry and
3 of grapes for a party of four children. If every child is allowed to choose a popsicle of his/her favourite flavour, what is the probability that all of them will get all their choices?
Ans: 26/27

How to show the answer? Any people help?
Thanks

 

[soroban]

Hello, tecklee89!

A mother prepares nine popsicles of different flavors:
3 Orange, 3 Cherry, and 3 Grape for a party of four children.
If every child is allowed to choose a popsicle of his/her favourite flavor,
what is the probability that all of them will get all their choices?

\(\displaystyle \text{Answer: }\:\tfrac{26}{27}\)

\(\displaystyle \text{Each of the four children could choose one of 3 flavors.}\)
. . \(\displaystyle \text{Hence, there are: }\:3^4 \:=\:81\text{ possible choices.}\)

\(\displaystyle \text{The only way that they will }not\text{ get their choices}\)
. . \(\displaystyle \text{is if all four children choose the same flavor.}\)
\(\displaystyle \text{This can happen in }three\text{ ways: }\:\{OOOO,\:CCCC,\:GGGG\}\)

\(\displaystyle \text{Hence, there are: }\:81 - 3 \:=\:78\text{ ways that they get their choices.}\)

\(\displaystyle \text{Therefore: }\(\text{get their choices}) \:=\:\frac{78}{81} \:=\:\frac{26}{27}\)