Question

Find the number of 6 letter words that can be formed using all letters of the English alphabet so that the words contain atleast 3 consonants and each letter of the word is unique in every case?

• A. (²¹C₃ x ⁵C₄ + ²¹C₄ x ⁵C₂+ ²¹C₅x ⁵C₁ + ²¹C₆)x 6!
• B. (²¹C₃ x ⁵C₃ + ²¹C₄ x ⁵C₂+ ²¹C₅x ⁵C₁ + ²¹C₆)x 6!
• C. ²¹C₃ x ⁵C₄ + ²¹C₄ x ⁵C₂+ ²¹C₅x ⁵C₁ + ²¹C₆
• D. none of these

Answer 1

The correct option is B

(²¹C₃ x ⁵C₃ + ²¹C₄ x ⁵C₂+ ²¹C₅x ⁵C₁ + ²¹C₆)x 6!

TOTAL = 21 consonants and 5 vowels
Three cases
3 consonants & 3 vowels= ${}^{21}$C${}_3$ $\times$ ${}^5$C${}_3$
4 consonants & 2 vowels= ${}^{21}$C${}_4$ $\times$ ${}^5$C${}_2$
5 consonants & 1 vowel= ${}^{21}$C${}_5$ $\times$ ${}^5$C${}_1$
6 consonants & 0 vowels= ${}^{21}$C${}_6$

Arrangement = 6! In each case

(${}^{21}$C${}_3$ $\times$ ${}^5$C${}_3$ + ${}^{21}$C${}_4$ $\times$ ${}^5$C${}_2$ + ${}^{21}$C${}_5$ $\times$ ${}^5$C${}_1$ + ${}^{21}$C${}_6$) $\times$ 6!