How many distinct permutations are there of the letters of the word bookkeeper? there are __[blank]__ distinct permutations of the letters of the word bookkeeper. a group of 3 english majors, 2 anthropology majors, and 5 history majors are going out to dinner, where they will sit at a circular table. if students with the same major need to sit together, how many different ways can the students be seated around the table?

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:


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