#### In how many ways can the letters of the word ‘ASSOCIATION’ be arranged such that all the consonants occur together?Option: 1 37800Option: 2 45500Option: 3 52300Option: 4 63600

In the word “ASSOCIATION” there are 11 letters.

Here, there are 6 vowels (i.e. 2 A’s, 2 O’s, and 2 I’s) and 5 consonants (i.e 2 S’s, and each of C, T, N)

Considering consonants as one letter, the number of letters becomes 7 which can be arranged is given by,

\begin{aligned} \frac{7 !}{2 ! 2 ! 2 !} & =\frac{7 \times 6 \times 5 \times 4 \times 3 \times 2}{2 \times 2 \times 2} \\ \frac{7 !}{2 ! 2 ! 2 !} & =630 \end{aligned}

Consonant S appears twice, so consonants can be arranged as

\begin{aligned} & \frac{5 !}{2 !}=\frac{5 \times 4 \times 3 \times 2}{2} \\ & \frac{5 !}{2 !}=60 \end{aligned}

Hence the required number of ways in which the letters of the word “ASSOCIATION” be arranged so that all the vowels occur together is given by,

\begin{aligned} 630 \times 60=37800 \end{aligned}

Therefore, the total number of ways to form the letter is 37800.