# Thread: 2 objects to be filled into 3 spaces

1. ## 2 objects to be filled into 3 spaces

The letters 'P, I, H, A, T ' are to be arranged in a row. Find the number of possible arrangement if the vocals letters occupy the even position

after filling in P, H , T into second and fourth location, now there's only 2 letter (I and A ) , to be filled into 2 locations. how to do this?

2. ## Re: 2 objects to be filled into 3 spaces

Originally Posted by delso
The letters 'P, I, H, A, T ' are to be arranged in a row. Find the number of possible arrangement if the vocals letters occupy the even position

after filling in P, H , T into second and fourth location, now there's only 2 letter (I and A ) , to be filled into 2 locations. how to do this?
I don't understand what you've written. You seem to be putting 3 letters into 2 slots.

Assuming the slots are 1-5 there are 2 even positions, 2 and 4.

There are only 2 ways to put the "vocals letters" (a better English word is vowels) into these slots. I=2, A=4; or A=2, I=4.

Then there are 3! ways of arranging the consonants (the remaining letters).

2*3! = 12

so there are 12 arrangements that have vowels in even numbered slots.

3. ## Re: 2 objects to be filled into 3 spaces

Hello, delso!

The letters 'P, I, H, A, T ' are to be arranged in a row.
Find the number of possible arrangement if the vowels occupy the even positions.

After filling in P, H, T into second and fourth location. . .
HOW did you fit three letters into two positions?

$\text{We have: }\;\_\;\_\;\_\;\_\;\_$

$\text{Place the two vowels: }\;\_\;\text{v}\;\_\;\text{v}\;\_$
There are $2!$ permutations of the vowels.

$\text{Place the three consonants: }\:\text{c}\;\text{v}\;\text{c}\;\text{v}\;\text{c }$
There are $3!$ permutations of the consonants.

Therefore, there are: . $(2!)(3!) \,=\,12$ possible arrangements.