In how many ways can the letters of word "EDUCATION" be arranged

BTGModeratorVI wrote: ↑

Mon Jul 13, 2020 7:42 am

In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!

Answer: C
Source: Math Revolution

VOWELS: E, U, A, I, O
CONSONANTS: D, C, T, N

Take the task of arranging the letters and break it into stages.

Stage 1: Arrange the 5 vowels
We can arrange n unique objects in n! ways
So, we can arrange the 5 vowels in 5! ways
In other words, we can complete stage 1 in 5! ways

Stage 2: Arrange the 4 consonants
There are 4 unique objects to arrange.
We can complete stage 2 in 4! ways

Stage 3: The two groups of letters (the vowel group and the consonant group)
There are two options:
1) CONSONANTS-VOWELS
1) VOWELS-CONSONANTS
We can complete stage 3 in 2 ways

By the Fundamental Counting Principle (FCP), we can complete the 3 stages (and thus arrange all 9 letters) in (5!)(4!)(2) ways.

Answer: D

Cheers,
Brent

BTGModeratorVI wrote: ↑

Mon Jul 13, 2020 7:42 am

In how many ways can the letters of word "EDUCATION" be arranged such that all the vowels always appear together?

A) 9!
B) 5!*4!
C) 5!*5!
D) 5!*4!*2!
E) 6!*4!

Answer: C

Solution:

We need to determine the number of ways to arrange the letters of the word "EDUCATION" such that all the vowels always appear together. We start by arranging the given word with the vowels together.

[EUAIO]-D-C-T-N

We think of [EUAIO] as a single letter; thus, there are 5! ways to arrange [EUAIO]-D-C-T-N.

Next we must determine the number of arrangements for the letters in [EUAIO].

[EUAIO] can be arranged in 5! ways.



Putting it together, we find that the total number of ways to arrange the letters in the word "EDUCATION" such that all the vowels always appear together is 5! x 5!.

Answer: C