Permutation and Combinations 4

Question

 How many arrangements of the letters of GRACIOUS are there with
the vowels occur consecutively (like GRCAOIUS, not GACRIOUS)? (Letters
do not need to be in alphabetical order)

Answer 1

you have 4 vowels, A, O, I, and U and they must occur consecutively

how many ways can you arrange the 4 vowels ?

first vowel you choose from 4

second vowel you choose from 3

third vowel you choose from 2

fourth vowel you choose from 1

4*3*2*1=24 ways (AOIU, AOUI, AUOI, AUIO, AIOU, AIUO, etc.)

how can you place them ?

C=consonant and V=vowel

VVVVCCCC and how many ways can you arrange G, R, C, and S ?

4*3*2*1=24 ways the consonants can be arranged

24*24=576 ways for VVVVCCCC

the consonants and vowels can be arranged also in the following ways:

CVVVVCCC  4*4*3*2*1*3*2*1=576 ways

CCVVVVCC  4*3*4*3*2*1*2*1=576 ways

CCCVVVVC  4*3*2*4*3*2*1*1=576 ways

CCCCVVVV  4*3*2*1*4*3*2*1=576 ways

you have 5 arrangements and each produces 576 ways

5*576=2880 arrangements with the 4 vowels occurring consecutively