Question

The number of words that can be formed from the letters of word ${}^{\prime }USAINBOL{T}^{\prime }$ whose middle place is a vowel, start with a vowel and end with a consonant is

• A. $36\cdot 6!$
• B. $12\cdot 6!$
• C. $60\cdot 6!$
• D. $18\cdot 6!$

Answer 1

The correct option is C $60\cdot 6!$

Given word : $USAINBOLT$
Here, vowels $=\{A,I,O,U\}$
Consonants $=\{S,N,B,L,T\}$

Let the word is $X_1{X}_2{X}_3{X}_4{X}_5{X}_6{X}_7{X}_8{X}_9$, then
$\left(i\right)$ Middle place$\left(X_5\right)$ can be filled in $4$ ways.
$\left(ii\right)$ The starting place $\left(X_1\right)$ can be filled in $3$ ways.
$\left(iii\right)$ The ending place $\left(X_9\right)$ can be filled in $5$ ways.

$\therefore$ The required ways $=4\cdot 3\cdot 5\cdot 6!=60\cdot 6!$