the first one 和 first one 都可以用，但有区别：

▲the first one 主要用于特指的场合。如：

Our house is the first one on the right. 我们的房子是右边的第一栋。

After their argument, he was the first one to hold out an olive branch. 争吵过后，他首先示好。

I was the first one to get there, Helen arrived after a few minutes. 我是第一个到达那儿的。几分钟后海伦就来了。

We had no idea which hotel would be best, so we just took pot luck with the first one on the list. 我们也不知道哪家旅馆最好，所以就碰运气挑了名单上的第一家。

I never thought you’d be the first one to walk down the aisle—you used to say you’d never marry! 我绝没有想到你会是第一个走向婚姻殿堂的人——你过去常说自己永远也不结婚!

▲ first one（没有定冠词），主要用于表示泛指的场合（通常与 another 连用）。如：

Twilight deepened, first one, then another star appeared. 暮色渐浓，先是一颗星星，接着又一颗星星出现了。（张道真《现代英语用法词典》第1卷第171页）

She chose first one, then another, but neither was to her satisfaction. 她先挑了第一个，接着又挑了一个，但两个她都不满意。（外研社•柯林斯《高级英语用法词典》第558页）

◎另外，当我们要表示两个当中“一个……另一个……”时，通常是用 one…the other…，但有时也说成 first one…the other…。比如英英词典对 backstroke(仰泳)的解释是：

Backstroke is a way of swimming on your back by moving first one arm then the other backwards while kicking your feet.