ratnanideepak wrote:
In how many ways can the letters of the word MANIFOLD be arranged so that the vowels are separated ?
A. 14400
B. 36000
C. 18000
D. 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: A
Note: the FCP can be used to solve the
MAJORITY of counting questions on the GMAT. So, be sure to learn it.
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