Question

How many words can be formed by arranging the letters of the word ‘MUMBAI’ so that all the M’s come together?

Answer 1

Hint – In this particular question the number of words formed is taking by two M’s as one and the remaining letters of the word MUMBAI so that there are 5 distinct words (4 remaining words + 1 word when two M’s take as one), so the arrangements of these letters is given as (n!), where n is the number of letters which we have to arrange so use these concepts to reach the solution of the question.

Complete step-by-step answer:
As we see in MUMBAI there are a total six letters but there is 2 M so there are 5 distinct letters.
Now we have to form the words such that all the M’s come together.
So consider the two M’s as one so there are 5 letters so we have to arrange them so the number of ways to form the words with or without meaning such that all the M’s come together is 5!
So the total number of words using the letters of MUMBAI such that all the M’s come together are 5!
Now as we know that n! = n (n – 1) (n – 2)...............(n – (n – 1))
So 5! = 5 (4) (3) (2) (1)
Now simplify this we have,
So 5! = 120
So the total number of words using the letters of MUMBAI such that all the M’s come together are 120.
So this is the required answer.

Note – We can also solve these types of problems using another method, consider the rest of the letters except M so there are four letters so the number of ways to arrange them is (4!). Now there are 3 places between these four letters and 1-1 places at the extreme left and at the extreme right position so the total number of spaces in which two M’s come together are (3 + 2) = 5 so the total number of words using the letters of MUMBAI such that all the M’s come together are the product of these vacant spaces and the arrangement of 4 letters = 5(4!) = 5(24) = 120.