a majority of和the majority of都可以表示大多数，区别只是在于不定冠词和定冠词，也就是说，a majority of表示泛指的大多数，the majority of表示特指的大多数。
这个问题涉及不定冠词和定冠词的用法：
英语中的冠词有三个，即不定冠词a、an和定冠词the。
一、不定冠词a、an的用法。
不定冠词a用在以辅音音素，包括半元音（不是辅音字母）开头的名词前，an用在以元音音素（不是元音字母）开头的名词前。如：
an honest girl
an hour
an heir
an honor
a useless tool
a university
a one-sided decision
a once chairman
a European country
a ewe
an A student
an H bomb
a VIP card
a BBC official
Is it with an o or is it with a u?
不定冠词的用法如下：
表示数量为一，一个的意思。
Rome was not built in a day.
I need a dozen eggs.
I have learned English for a score years.
表示泛指一类人或物。
A triangle has three angles.
A skunk is a smelly animal.
A knife is a tool with which you cut things.
A family is a group of people related by blood or marriage.
用在专有名词前面，表示某一个，某位，或某家庭的成员，或某产品的型号。
A Buick is a car with dual exhaust pipes.
His wife is a Stewart.
A Mr. Robertson wanted to see you.
Never would he think of such a prosperous New York.
He came back on a Christmas.
表示“每一”，如：
How much a dollar is it?
They cost only one dollar a dozen.
How much is this cloth a yard?
He drove at forty miles an hour.
二、定冠词the的用法。
1. 表示独一无二的事物的名词前。
The earth spins around the sun.
The moon turns around the earth.
The sun rises in the east.
The bank is on the right.
2. 表示特指的人或物的名词前。
Do you remember to lock the door?
Do you have the key to the trunk?
The wine I drank at the bar was very good.
I have 21 customers, the majority of whom can afford it.
3. 形容词最高级，副词最高级或序数词、only、very前。
He is the first man to come.
He is much the best boy of all.
Is this the very first aerobics class that you attended?
4. 用在形容词前，表示一类人或事物。
The wounded numbered ten.
The rich aren't always happy.
5. 乐器名称前面。
You play the violin with a bow.
You play the flute on the lips.
6. 表示单位的名词前。
We were hired by the hour.
Meat is sold by the pound.
Coal is sold by the ton.
Milk is sold by the pint.
7. 定冠词可以用以表示对人体部位的特指，如：
He shook her roughly by the shoulder.
He hit the girl on the nose.
The man was shot through the heart.
Richard hit Robert in the face.
She kissed her dad on the cheek.