my way of solving the permutation question

Question

In how many ways can all the letters in the word TATTOO be arranged if the vowels must be together in the order AOO?

my work:

in the word TATTO there are 6 letters

so 6p6

=6!/(6-6)! =6!/0! =6! =6x5x4x3x2x1=720

answer=720

Answer 1

Hint : AOO are together so consider them together as a single letter, say $AOO=\lambda$ now you have to find ways to arrange $T,T,T, \lambda$

Formula

Ways to arrange $n$ objects when $m$ are same. $$\frac{n!}{m!}$$

Answer 2

If we force AOO to be together, we can simply replace them with a single letter, say X. Then there are only four possibilities, depending on the place of X among the Ts.

Answer 3

That's a wrong way brother....Here's how to do it! we've the word TATTOO and we've to arrange its letters in such a way that vowels are together in the order (AOO),so we can rearrange the word for our convenience in the following manner, (AOO)TTT. Now we can treat (AOO) as a single entity....say 'x' So,we've (AOO)TTT = xTTT. Now total no. of arrangements can be calculated using the formula for 'Total no of arrangements of n objects having 'p' objects of one kind, 'q' objects of second kind and 'r' objects of third kind i.e.,

P = n!/p!.q!.r! So in the above case,

P = 4!/3! = 4.3!/3! = 4 Hence, total number of arrangements are "4"...Hope this helped!