The following question has three statements. Study the question and the statements and decide which of the statement(s) is necessary to answer the question. An 8-letter word has no repetitive letter. Find the number of vowels in the word. I) No. of ways in which the consonants can be arranged is 720. II) No. of ways in which the vowels can be arranged is 2. III) 10080 different words can be formed keeping the vowels together.?

Option 3 : Any one

Let the no. of vowels be ‘x’

No. of consonants = (8 - x)

Considering statement I,

No. of ways in which consonants can be arranged = (8 - x) ! = 720

Putting x = 1, 7! = 5040 ≠ 720

Putting x = 2, 6! = 720

Hence, no. of vowels = 2

Considering statement II,

No. of ways in which vowels can be arranged = x! = 2

Putting x = 1, 1! ≠ 2

Putting x = 2, 2! = 2

Hence, no. of vowels = 2

Considering statement III,

When x vowels come together, no. of ways of arranging them = x!

Total no. of letters to arrange = 8 - x + 1 = (9 - x)

No. of ways of arranging them = (9 - x) !

No. of different words = x! × (9 - x) ! = 10080

Putting x = 1, 1! × 8! = 40320 ≠ 10080

Putting x = 2, 2! × 7! = 10080

Hence, no. of vowels = 2