From a set of c distinct consonants and v distinct vowels, 5 -letter words are formed using exactly 3 consonants and 2 vowels. If the number of such words where the two vowels are always adjacent is 16800 , then the total number of letters in the original set is

Step-by-step solution

First, we select 3 consonants from c distinct consonants and 2 vowels from v distinct vowels. Number of ways to select = ^ c C₃ ^ v C₂ For the arrangement, the 2 selected vowels must be adjacent. We treat them as a single block. This block along with the 3 consonants gives 4 entities to arrange, which can be done in 4! ways. The 2 vowels can be arranged among themselves in 2! ways. Total number of words = ^ c C₃ ^ v C₂ 4! 2! = 16800 ^ c C₃ ^ v C₂ 24 2 = 16800 ^ c C₃ ^ v C₂ = 350 c(c-1)(c-2) 6 v(v-1) 2 = 350 c(c-1)(