In how many ways can the letters of the word ‘STRANGE’ be arranged

1.	In how many ways can the letters of the word ‘STRANGE’ be arranged so that the vowels come together?
Answer» Given : The word is ‘STRANGE.’ To find : A number of arrangements in which vowels come together Number of vowels in this word = 2(A, E) Now, Consider these two vowels as one entity (AE together as a single letter) So, the total number of letters = 6(AE S T R N G) Formula used : Number of arrangements of n things taken all at a time = P(n, n) P(n, r) = \(\frac{n!}{(n-r)!}\) ∴ Total number of arrangements = the number of arrangements of 6 things taken all at a time = P(6, 6) = \(\frac{6!}{(6-6)!}\) = \(\frac{6!}{0!}\) {∵ 0! = 1} = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 Two vowels which are together as a letter can be arranged in 2 Ways like EA or AE Hence, Total number of arrangements in which vowels come together = 2 × 720 = 1440

In how many ways can the letters of the word ‘STRANGE’ be arranged so that the vowels come together?

Answer»

Given : The word is ‘STRANGE.’

To find : A number of arrangements in which vowels come together

Number of vowels in this word = 2(A, E)

Now,

Consider these two vowels as one entity

(AE together as a single letter)

So, the total number of letters = 6(AE S T R N G)

Formula used :

Number of arrangements of n things taken all at a time = P(n, n)

P(n, r) = \(\frac{n!}{(n-r)!}\)

∴ Total number of arrangements

= the number of arrangements of 6 things taken all at a time

= P(6, 6)

= \(\frac{6!}{(6-6)!}\)

= \(\frac{6!}{0!}\)

{∵ 0! = 1}

= 6!

= 6 × 5 × 4 × 3 × 2 × 1

= 720

Two vowels which are together as a letter can be arranged in 2

Ways like EA or AE

Hence,

Total number of arrangements in which vowels come together = 2 × 720 = 1440