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

In the word 'STRANGE', there are 2 vowels ('A' and 'E') and 5 consonants ("S", "T", "R", "N" and "G").

(i)
Bind the two vowels together as a group.
Internal arrangement of the group of the two vowel is ₂P₁.
Arrange the 5 consonants and the group of the two vowels (₆P₆).

Number of ways of arrangement
= ₂P₁ × ₆P₆
= 2 × 6!
= 1440

(ii)
Arrange the 5 consonants as : _X_X_X_X_X_ (₅P₅)
where 'X' is a consonant, and '_' is the space between two consonants.
Among the 6 '_', put the two vowels in 2 '_'. (₆P₂).

Number of ways of arrangement
= ₅P₅ × ₆P₂
= 5! × (6 × 5)
= 3600

(iii)
Out of the 5 consonants, choose 3 to arrange as : _X_X_X_ (₅P₃)
where 'X' is a consonant, and '_' is the space between two consonants.
Put the 2 vowels and the rest 2 consonants into the 4 "_". (₄P₄)

Number of ways of arrangement
= ₅P₃ × ₄P₄
= (5 × 4× 3) × 4!
= 1440