We have $n$ colored pebbles, where each pebble is either red or green or

blue. We want to order the pebbles in a straight line of length $n$ such

that no two adjacent pebbles will have the same color. How many

possibilities do we have to do so$?$

$($For example: A valid ordering is "red blue red blue red blue $...",$ and a

non$-$valid ordering is "red red blue red blue red blue $...",$ because the

first two pebbles are red.$)$

$($Hint: This question is identical to a question asking about the number

of strings of length $n$ with elements from $[3]$ such that no two adjacent

elements in the string are identical.$)$

Let $A$ be a set of size $n$ and $B$ be a set of size $m,$ and denote $r=|AB|.$

Alice sits in a room alone and chooses an element from A$,$ and Bob sits

in a different room alone and chooses an element from B$.$ Then they

meet, compare their choices, and if their choices are identical they

celebrate. How many different pairs of choices do not lead to a

celebration?

$($To assist in parsing the question: A pair of choices consists of two

elements, the element chosen by Alice and the element chosen by Bob.

For example, if Alice chose xinA and Bob chose yinB, then they chose

the pair $(x,y)$; this pair led to a celebration if and only if $x=y.)$

There are $n$ different restaurants at the UTSC campus, and we need to

choose a subset of the best restaurants. Alice claims that we should

select $n-knnk$ restaurants, and Bob claims that $we$ should select $n-k$

restaurants. Would the number $of$ possible choices $be$ greater $ifwe$

choose $k$ restaurants, $orifwe$ choose $n-k$ restaurants?

What other name $dowe$ have for $ann-$permutation $ofn$ elements?