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In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated?
I tried total permutations in which vowels are together, which gives 36000 which was wrong.
$\endgroup$ 1 Re: In how many ways can the letters of the word MANIFOLD be arranged so t [#permalink]
Carcass wrote:
In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ?A. 14400B. 36000C. 18000D. 24000
E. 22200
Take the task of arranging the 8 letters and break it into stages.
Stage 1: Arrange the 5 CONSONANTS (M, N, F, L and D) in a row
We can arrange n unique objects in n! ways. So, we can arrange the 5 consonants in 5! ways (= 120 ways)So, we can complete stage 1 in 120 ways
IMPORTANT: For each arrangement of 5 consonants, there are 6 spaces where the VOWELS can be placed. For example, in the arrangement MNDLF, we can add spaces as follows _M_N_D_L_F_So, if we place each vowel in one of the available spaces, we can ENSURE that the vowels are separated.Stage 2: Select a space to place the A.
There are 6 spaces to choose from, so we can complete stage 2 in 6 ways.
Stage 3: Select a space to place the I.
There are 5 remaining spaces to choose from, so we can complete stage 3 in 5 ways.
Stage 4: Select a space to place the O.
There are 4 remaining spaces to choose from, so we can complete stage 4 in 4 ways.
By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus arrange all 8 letters) in (120)(6)(5)(4) ways (= 14,400 ways)
Answer: ANote: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.
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