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Find the number of ways in which the letters of the word “machine” can
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26 Oct 2021, 05:04
26 Oct 2021, 05:04
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23% (01:42) correct
76% (01:51) wrong based on 13 sessions
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Find the number of ways in which the letters of the word “machine” can be arranged such that the vowels may occupy only odd positions ?
Show: ::
576
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Re: Find the number of ways in which the letters of the word “machine” can
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26 Oct 2021, 05:28
26 Oct 2021, 05:28
1
Carcass wrote:
Find the number of ways in which the letters of the word “machine” can be arranged such that the vowels may occupy only odd positions ?
Show: ::
576
There are 7 letters in the word MACHINE
There are 3 vowels: A, I and E
The odd positions are the 1st, 3rd, 5th and 7th positions.
We’ll begin with the most restrictive stage.
Stage 1: Select a position to place the A
There are 4 ODD positions available (the 1st, 3rd, 5th and 7th positions)
So, we can complete stage 1 in 4 ways
Stage 2: Select a position to place the I
After Stage 1 is completed, there are 3 ODD positions remaining.
So, we can complete stage 2 in 3 ways
Stage 3: Select a position to place the E
After Stages 1 and 2 are completed, there are 2 ODD positions remaining.
So, we can complete stage 3 in 2 ways
Stage 4: Select a position to place the M
There are now 4 spaces remaining. So, we can complete stage 4 in 4 ways
Stage 5: Select a position to place the C
There are now 3 spaces remaining. So, we can complete stage 5 in 3 ways
Stage 6: Select a position to place the H
There are now 2 spaces remaining. So, we can complete stage 6 in 2 ways
Stage 7: Select a position to place the N
There is now 1 space remaining. So, we can complete stage 7 in 1 way
By the Fundamental Counting Principle (FCP), we can complete all 7 stages (and thus place al 7 letters) in (4)(3)(2)(4)(3)(2)(1) ways (= 576 ways)
Answer: 576
Note: 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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General Discussion
Re: Find the number of ways in which the letters of the word machine can
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02 Dec 2022, 17:56
02 Dec 2022, 17:56
1
Is this even possible to compute since there are only 3 vowels and four odd spots in the letter combinations of the word "machine"?
