Exercise 6. Determine the number of anagrams of the word PROBLEME, meaningful or not, such that two E letters do not follow each other. (For example, the anagram RPOELBEM is valid but not POREEBML.)
Solution to exercise 6 We start by permuting the letters PROBLM: this gives $1 \times$ $2 \times \cdots 6=720$ possibilities. Then, we need to place two E's in 7 positions, which gives another 21 choices. The answer is 15120.