Question

Find the number of words each containing of $3$ consonants and $3$ vowels that can be formed from letters of the word Circumference. In how many of these all c's will be together.

Answer 1

Given Word - Circumference

Consonants - {c,c,c},{r,r},m,n,f

Vowels - i,u,{e,e,e}

Total no. of ways of selecting and arranging 3 consonants = {3 Alike}+{2 Alike + 1 Distinct}+{3 Distinct}

=1+${}^2{C}_1{\times }^4{C}_1\times \frac{3!}{2!}{+}^5{C}_3\times 3!$=85

Total no. of ways of selecting and arranging 3 vowels = {3 Alike }+{2 Alike 1 Distinct}+{3 Distinct }

=1+$1{\times }^2{C}_1\times \frac{3!}{2!}$+${}^3{C}_3\times 3!$=13

Total no. of ways of arranging them together i.e total no. of words that can be formed = ${}^6{C}_3\times 85\times 13=20\times 85\times 13=22100$

Let's see the case of all c's together now ,

Total no. of ways of selecting consonants = 1

Total no. of ways of selecting and arranging vowels =13

Total no. of words = ${}^4{C}_1\times 13=52$