running out - to "run out" out of something is to deplete your stock of that item (usually quickly)
selling out - when something is "selling out," this means that they are starting to become "sold out," or that they are so many sold that soon there will be none left
running out - to "run out" out of something（何かを切らす）というのは、商品の在庫が（通常早く）なくなることです。
selling out - "selling out"というのは、"sold out"（売り切れ）になりつつあること、あるいはたくさん売れてしまったのでもうすぐなくなることを意味しています。
The phrase 'remaining seats' means 'available seats' or 'vacant seats left'. All these phrases can be used when referring to the decreasing number of seats on the plane to your destination. You may also refer to the decreasing number of seats as 'becoming fewer and fewer'. The adjective 'few' means little in number. In this case, the comparative degree, 'fewer' has been used. You may the adjective 'vacant' to mean empty.
So, you may say:
The remaining seats are decreasing very fast.
or
The available seats are becoming fewer and fewer.
or
There are very few vacant seats left.
'remaining seats' とは、'available seats'（可能な席）または 'vacant seats left'（空いている席）を意味します。目的地への飛行機の残席数のことをいいたいときにこれら全て使うことができます。
残席数を'becoming fewer and fewer'（だんだん少なくなってきている）と表すことができます。 形容詞'few' は、数が少ないという意味です。この場合、比較級 'fewer'が使われています。 形容詞 'vacant'は、空のという意味です。
The remaining seats are decreasing very fast.
残席数がとてもはやく減ってきている。
The available seats are becoming fewer and fewer.
空席はだんだん少なくなってきている。
There are very few vacant seats left.
空席がほんの少しだけ残っている。
It seems like the flight tickets are selling fast.
The available seats are running out quickly.
*It seems like the flight tickets are selling fast.
Remember people do not purchase the seats but the fight tickets. Therefore if the tickets are selling fast it means the flight seats are also running out quickly.
*The available seats are running out quickly.
Available seats means the seats you can select/chose from.
例文
*It seems like the flight tickets are selling fast.
航空券はあっという間に売れている。
席ではなく、航空券を買うことに気をつけないといけません。
ですからチケットがあっという間に売れているということは、
航空券がすぐに売り切れるということです。
例文
*The available seats are running out quickly.
購入可能な席は すぐになくなる。
Available seatsは、購入可能な席ということです。
At a premium = scarce and in demand, scarce, in great demand, like gold dust, hard to come by, in short supply, thin on the ground, few and far between, not to be had, rare, rare/scarce as hen's teeth; informal: not to be had for love or money
"Office space in the city was at a premium."
"Parking space is at a premium in Japanese cities."
At this rate = if matters continue in this or that way.
"At this rate, I won't have a job to go back to."
At a premium = 希少で需要がある、希少な、大きな需要がある、砂金のような、供給が不十分な、ごく稀な、ない、稀な、稀な/とても希少な；
カジュアルな表現：金ずくでは買えぬ
【例文】
"Office space in the city was at a premium."（市街のオフィススペースはなかなかない）
"Parking space is at a premium in Japanese cities."（日本の街では駐車スペースが希少だ）
At this rate = if matters continue in this or that way.（このまま事が続けば）
"At this rate, I won't have a job to go back to."（このままだと戻れる仕事がなくなる）
There aren't many seats left on the flight we want to get
Seats for the flight are running out
When you're trying to explain that flight seats are getting less and less, then you can say:
-There aren't many seats left on the flight we want to get.
-Seats for the flight are running out.
飛行機の座席が少なくなってきていることを説明するなら、次のように言えます。
-There aren't many seats left on the flight we want to get.
私たちが乗りたいフライトの座席は、もうあまり残ってません。
-Seats for the flight are running out.
このフライトの座席が、残り少ないです。
"Most seats are filled and There are hardly any seats left"
Both of these phrases are appropriate to express that there are no seats left or that there are a few left. In this case it just comes down to preference in what to say since both express the same idea. Some people may even say "Seats are filling up quickly lets be quick about selecting the seats.
"Most seats are filled"
「ほとんどの席が埋まっています」
"There are hardly any seats left"
「ほとんど席は残っていません」
二例とも、残席が（ほとんど）ない場合に使えるフレーズです。どちらを使うかは好みの問題です。どちらも同じことを表しています。
また、他に：
"Seats are filling up quickly. Let's be quick about selecting the seats."
「席がどんどん埋まっています。早く席を選びましょう」
と言うこともあるかもしれません。
アメリカ合衆国 
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