Question

How many arrangements of the letters in TOMATO are there, if the letters O are to be separated?

Answer 1

it means you can't have OOTMAT but you can have TTOMAO right?

Answer 2

let's find the total no. of words that can be formed using tomato
T=2
O=2
M=1
A=1
total no. of letters=6
so no. of words that can be formed=6!/(2!*2!)=180
now let's find the no. of arrangements when both Os are together
then we'll have 5 letters, considering both Os together
5!/2!=120/2=60
to find the no. of arrangements when Os are separate=180-60=120

Answer 3

if yes
1 part:
then when O is first or last we have:
1 _ _ _ _ _
we can't use O and another O so 4 left
1 4 _ _ _ _
we can use all remaining letter
1 4 4 3 2 1
we double counted T so need to divide by 2!=2
and we also need to multiply by 2 because same thing works when O is last letter
1*4*4*3*2*1/2*2
2 part:
when O is any position between 2nd and 5th
_ 1 _ _ _ _
letters to O can't be O itself
4 1 _ _ _ _
4 1 3 _ _ _
and now we can use all letters
4 1 3 3 2 1
we double counted T so need to divide by 2
and O can start in 2nd, 3rd, 4th, 5th position so we have 4 same scenarios so need to multiply by 4
4*1*3*3*2*1/2*4
3 part:
add first and second part!

(((1*4*4*3*2*1)/2)*2)+(((4*1*3*3*2*1)/2)*4)=240

Answer 4

which answer is correct one? :D

Answer 5

I too got 240, though I am not sure.

Answer 6

how you have done it?

Answer 7

O 4 O 3 2 1 (O is alphabet O, not zero) 4 times
O 4 3 O 2 1 3 times
O 4 3 2 O 1 2 times
O 4 3 2 1 O 1 time
24*10=240

Answer 8

Any mistake ?

Answer 9

there are 2 Ts, so that has to be divided by 2!

Answer 10

But how does that 2 T's affect in any way ?

Answer 11

let's have word 'att'
if you don't consider the two ts and treat them as different letters
you'll have arrangements repeated
tat, tat, att, att, tta, tta
following this way we'll get 3! words=6
but actually there are only 3 distinct arrangements
att,tat,tta
i.e 3!/2!=3

Answer 12

yeah you need to divide by 2

Answer 13

Oh, Ok. Then your answer must be right.

Answer 14

but why my answer incorrect then ><

Answer 15

ash you used 6C4 in first calculation?

Answer 16

no. i didn't use it

Answer 17

you just eliminated double counting and it coincidence that it's same formula? right? :D

Answer 18

yeah:D