Question

Find the number of 6 letter words that can be formed using all letters of the English alphabet so that the words contain at least 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!