Question: You have 9 English books, 7 French books and 5 German books. How many ways are there to make a row of three books in which exactly one language is missing (the order of the three books makes a difference) ?
I do not want to use permutations (nPr). My solution is: $\left(\binom{16}{3} + \binom{14}{3} + \binom{12}{3}\right)\times 3$ My logic is: pick 3 books out of English & French (16) + pick 3 books out of English & German (14) + pick 3 books out of French & German (12) and multiply by 3 since the order matters (and there are three slots).
Is my answer correct?
Thanks!