Question

How many arrangements of the letters in INSTRUCTOR have all of the following properties simultaneously? (a) The vowels appearing in alphabetical order (b) At least 2 consonants between each vowel (c) Begin or end with the 2 Ts (the Ts are consecutive)

Answer 1

Answer:

480

Explanation:

INSTRUCTOR

This is:-

I,

N

S

T T

R R

U

C

O

Vowels are - I, O, U

They are written in alphabetical order.

Both T's have to be in start or end.

Also, there must be 2 consonants between each vowel

So,

Case:1 - T T ... I _  _ ... O _ _ .. U ...

Where, ...  represents the available spaces (4)

_ is the fixed spots

Case:2-     .. I _  _ ... O _ _ .. U ...T T

Thus, $2\times 4\times \frac{5!}{2!}=480$

2 is as there are 2 cases

4 is because 4 available spaces and one can go each time

$\frac{5!}{2!}=480$ for rest alphabets in which R appear 2 times