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Bunuel
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In how many ways can the letters of the word MACHINE be arranged so 01 Aug 2022, 02:20
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

65% (hard)

Question Stats:

61% (02:01) correct 39% (02:03) wrong based on 283 sessions

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In how many ways can the letters of the word MACHINE be arranged so that the vowels may occupy only odd position?

(A) 20160
(B) 576
(C) 288
(D) 144
(E) 72


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B
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In how many ways can the letters of the word MACHINE be arranged so 01 Aug 2022, 03:23
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Solution:

M A C H I N E

3 vowels - A I E
_1_ _2_ _3_ _4_ _5_ _6_ _7_

1] arranging vowels at 4 odd places = 4C3 X 3!
2] arrange the consonants = 4!

AND operator used

= 4C3 X 3! X 4!
=576
Option B
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GMAT 1: 760 Q50 V42
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In how many ways can the letters of the word MACHINE be arranged so 01 Aug 2022, 04:54
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Total vowels = 3(A, I, and E)

Total words = 7

Thus, there are 4 odd places.

3 vowels can be arranged on 4 odd places in 4*3! ways = 24

4 consonants can be arranged in 4! ways.

Thus, total ways = 4!*24 = 576

Thus the correct answer is B.
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In how many ways can the letters of the word MACHINE be arranged so 01 Aug 2022, 08:02
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Bunuel
In how many ways can the letters of the word MACHINE be arranged so that the vowels may occupy only odd position?

(A) 20160
(B) 576
(C) 288
(D) 144
(E) 72

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Step 1: Figure out which three of the four odd slots will have a vowel. There are four ways to select which one won't have a vowel, so 4.
Step 2: Figure out what order to place the three vowels in those three slots, so 3*2*1.
Step 3: Figure out what order to place the four consonants in the four remaining slots, so 4*3*2*1.
4*3*2*1*4*3*2*1 = 576.

Answer choice B.
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In how many ways can the letters of the word MACHINE be arranged so 16 Aug 2022, 08:17
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Permutations for 3 vowels and 4 consonants
3*4*2*3*1*2*1 = 144

Need to pick 3 positions out of 4 positions for the vowels, so we use combination 4C3 = 4

144*4=576

Posted from my mobile device
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In how many ways can the letters of the word MACHINE be arranged so 07 Nov 2025, 09:18
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if we number the positions from 1 to 7, we have 4 odd places.
if we number the positions from 0 to 6, we have 3 odd places.

so, we have 2 cases to think about. But how are we concluding that the numbering is done from 1 to7.
please help me to clear this confusion
Bunuel
In how many ways can the letters of the word MACHINE be arranged so that the vowels may occupy only odd position?

(A) 20160
(B) 576
(C) 288
(D) 144
(E) 72


Are You Up For the Challenge: 700 Level Questions­
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