Question

In how many ways the letters of the word rainbow must be together

Answer 1

Let all consider the word RAINBOW. It has 7 letters with no repetition which has:

No: of consonant letters = 4(i.e. r, n, b alsnd w)

No: of vowel letters = 3(i.e. a, i and o)

The question is vowels shouldn’t come together which is just opposite of vowels should come together. So, first we should find all the ways of arranging letters of RAINBOW and then subtract it by all the ways of arranging letters when vowels come together.

Let’s do it practically;

No: of ways in which all letters of RAINBOW can be arranged = 7! = 5040

Now,

No: of ways in which all letters of RAINBOW can be arranged when vowels come together: (It is little complicated one. Pay attention!)

When vowels come together(i.e. a, i and o) , we can consider all vowels as one single letter. So, there are 5 letters now (4 consonants + 1(consisting all vowels letter a, i and o) which can be arranged in:

= 5! = 120 ways

also,

the 3 vowels letters can be arranged themselves in = 3! = 6 ways

So, the total no: of ways the word ‘RAINBOW’ can be arranged is = 120 * 6 = 720 ways

Hence, the final answer in which vowels should not come together:

= all ways of arranging letters of RAINBOW - all ways of arranging letters of RAINBOW when vowels come together

= 5040 - 720

= 4320 ways

The concept is simple here:

all the ways of arranging letters of RAINBOW = all the ways with vowels always together + all the ways without vowels together.

NOTE:

The answer given below might seem convincing first. But we should know the word RAINBOW has 7 letters not 9 letters.

If you like my answer mark as BRAINLIST and follow me.

Answer 2

7 letters 'rainbow' has. And no letters are repeated.

So the answer is 7! = 5040.