Form sevenletter word by mixing up the letters in the word F

Form seven-letter word by mixing up the letters in the word FIXTURE. How many ways can you do this? How many ways can you do this if all the vowels have to be at the beginning? How many ways can you do this if no vowel is isolated between two consonants?

Solution

a>

Let say repeatition is not allowed then :
the first place could be filled by 7 letters
the second place could be filled by 6 letters
and so on
so total ways = 7*6*5*4*3*2*1 = 7! = 5040

b> in the word FIXTURE
total vowels= {I , U , E}= 3
total consonents = {F , x , T , R} = 4

if all the vowels need to be in the begining then
the first three places would be filled in = 3*2*1 ways= 6 ways
and the next four places would be filled by the consonents in = 4*3*2*1 = 24 ways

hence total ways = 6*24 = 144 ways