The letters of the word BOOKKEEPER can be arranged in 151,200 ways.
The group can be arranged in 8640 ways with all students of the same major together.
Explanation
There are 10 letters in the word bookkeeper. There is 1 B; 2 O's; 2 K's; 3 E's; 1 P; and 1 R.
An arrangement of n total objects where n₁ is one kind, n₂ is another, etc. is given by:
[tex]\frac{n!}{n_1!\times n_2!\times ... \times n_k!}
\\
\\=\frac{10!}{1!2!2!3!1!1!} = \frac{10!}{2!2!3!} = 151,200[/tex]
Keeping all of the students of each major together makes each one essentially a "unit." With this in mind, there are 3 units, that can be arranged in 3!=6 ways.
Within the English unit, the students can be arranged in 3!=6 ways.
Within the anthropology unit, the students can be arranged in 2!=2 ways.
Within the history unit, the students can be arranged in 5!=120 ways.
This gives us 6(6*2*120) = 8640
