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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.
Re: Find the number of ways in which the letters of the word machine can
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25 Mar 2023, 03:36
1
There are 3 vowels: A, I, E. The total number of letters for the word MACHINE is 7 which means the number of odd places is 4. So, the number of ways A can be in odd places = 4, for E = 3, and for I = 2, which implies the number of ways AIE can be arranged = (4)(3)(2) = 24 Also, the number of ways M, C, H, N can be arranged in the remaining places after vowels take 3 odd positions = (4)(3)(2)(1) = 4! = 24 Therefore, the total number of arrangements of the letters = 24 x 24 = 576
Re: Find the number of ways in which the letters of the word machine can
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02 Jun 2024, 16:54
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Re: Find the number of ways in which the letters of the word machine can [#permalink]