O, I, E should not come together.

-P-R-M-S-

There are five positions to be filled by 3 letters. Number of selecting 3 positions for the vowels is 5C3.
The vowels can inturn rearrange among themselves in 3! ways.

The consonants P, R, M, S can also arrange among themselves in 4! ways

so the total number of ways: 4! *3!* 5C3
=>1440 ways.

shashank.ism wrote:The letters of the word PROMISE are arranged so that no two of the vowels should come together. Find total number of arrangements.

a) 49
b) 1440
c) 7
d) 1898
e) 2005

Fibonacci, thanks a lot and I totally agree with your approach.
The way I approached was to put all vowels together and subtract it with total # of ways.
PROMISE can be arranged in factorial 7 ways. i.e 5040
Now lts consider OIE as one word, then PRMSV (where V = OIE) can be arranged in factorial 5 ways i.e 120
also OIE can be arranged in factorial 3 days i.e 6 ways.
Total PMRSV = 120*6 = 720 ways
Now # of ways where vowels CANNOT be together is = Total # of ways - # of ways where vowels are together
5440-720 = 4720 which is not one of the options.
Any light where I am taking it in wrong direction?

hey tata,
your approach takes care of situations where in all 3 vowels do not lie together, but what about 2 vowels together and 1 placed far off?

The method does not take this factor into account.

got it???

amount of times the rule is satisfied:10x3!x4! or 1440

b/c there are 10 different position types for the vowels not to be next to each other (listed below)
3! different combinations of o,i and e
4! combinations of everything else


v.v.v..
v.v..v.
v.v...v
v..v.v.
v..v..v
v...v.v
.v.v.v.
.v.v..v
.v..v.v
..v.v.v

fibbonnaci wrote:hey tata,
your approach takes care of situations where in all 3 vowels do not lie together, but what about 2 vowels together and 1 placed far off?

The method does not take this factor into account.

got it???

Agreed, thanks a lot for pointing that out Fibonacci.