Question

The number of ways in which the letters of the word "VALEDICTORY" be arranged so that the vowels may never be separated is

Answer 1

Here we need to find the number of vowels present in the word "VALEDICTORY". So we will first count the number of the words present in the given word. As the vowels can never be separated while arranging the letters so we will consider all the vowels together as one. We will then arrange the rest of the letters by taking the vowels as a single entity.

Complete step-by-step answer:
The given word is "VALEDICTORY” and there are 4 vowels in the word "VALEDICTORY” and those are A, E, I, O.
Now, we will assume that all these vowels are a single entity.
We can clearly see that the total number of letters in the word "VALEDICTORY” without taking vowels as one is equal to 11.
After taking vowels as 1, the total number of letters $= \left( {11 - 4} \right) + 1 = 7 + 1 = 8$
Total number of letters $= 8$.
Total number of ways to arrange these letters in a word $= 8!$
Now, we may also observe that as we are taking the pack of vowels as one entity then the letters within that entity will also arrange.
There are 4 vowels within the vowel entity, so they will arrange in $4!$ ways.
The required number of ways to arrange the letters in a word $= 8! \times 4!$
Computing the factorial, we get
The required number of ways to arrange the letters in a word $= 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 4 \times 3 \times 2 \times 1$
On multiplying the numbers, we get
The required number of ways to arrange the letters in a word $= 967680$

Note: In order to solve this question , we need to know the terms vowels and consonants. There are 26 alphabets in which 5 are vowels I.e. A,E,I, O and U and the rest 21 alphabets are consonants. Here, we have taken vowels as a single entity because we had to arrange the letters in such a way that it cannot get separated. If we don’t consider it as a single identity then the vowels will be arranged in any way, may be together or separated. This will violate the given condition and thus we will get wrong answers.