Finding the Number of Words with Fixed Vowel Order

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Homework Help Overview

The problem involves finding the number of distinct arrangements of the letters in the word "MATHEMATICS," specifically focusing on the vowels A, E, A, I, which must appear in that order. The remaining letters are consonants, and the challenge is to count the arrangements while adhering to the vowel order constraint.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the initial approach of calculating total arrangements without restrictions and the subsequent realization of the need to account for consonant placements. Questions arise regarding the correct method to fix vowel positions and how to effectively count arrangements.

Discussion Status

Some participants have offered guidance on simplifying the counting of vowel placements and suggested alternative methods for arranging consonants. There is an acknowledgment of confusion and a lack of consensus on the best approach, with various interpretations being explored.

Contextual Notes

Participants express uncertainty about the fixed positions of vowels and the implications of their order on the overall arrangement. There is also mention of potential constraints related to the problem setup and the need for clarity on how to approach the counting of arrangements.

ritwik06
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Homework Statement


Consider the word MATHEMATICS. There are some vowels: AEAI
The remaining 7 letters are MTHMTCS. Find the number of different 11 lettered words formed from these particular letters (repetition not allowed) such that all the vowels occur in the same order AEAI.
For example:
SCAHTEAIMMT



The Attempt at a Solution


The total number of words that can be formed is 11! without any restrictions.
If we fix the A at place 1:
no. of combinations of 3 from remaining: 10C3
If we fix the A at place 2:
no. of combinations of 3 from remaining: 9C3
If we fix the A at place 3:
no. of combinations of 3 from remaining: 8C3
...
...
Therefore the number of arrangements should be: 10C3+9C3+8C3+...+3C3
=330
which gives me a wrong answer. Why? What should I do to get a correct answer?
 
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You aren't accounting for the ways to place the consonants after the vowels are fixed. You are also counting the vowel placements the hard way. Why not just 11C4? Who cares where 'A' is??
 
Dick said:
You aren't accounting for the ways to place the consonants after the vowels are fixed. You are also counting the vowel placements the hard way. Why not just 11C4? Who cares where 'A' is??

Ok, I admit I am wrong. I am completely stuck with the question. Please help me out!
 
Its seems a bit hard to me. Nothing is fixed except the order of the vowels. What shall I do to this. Assuming seven places between each vowel doesn't help. I have also tried to fix the consonants first. Help :smile:
regards
 
Last edited:
You AREN'T wrong so far. You just aren't finished. You counted the vowel placements correctly (though as, I say, the hard way). So take one of your vowel arrangements. There are seven empty spaces left and you have seven consonants to put in them. How many ways can you do that?? The number of consonant arrangements doesn't depend on the particular vowel arrangement, right? So you can just multiply them.
 
Dick said:
You AREN'T wrong so far. You just aren't finished. You counted the vowel placements correctly (though as, I say, the hard way). So take one of your vowel arrangements. There are seven empty spaces left and you have seven consonants to put in them. How many ways can you do that?? The number of consonant arrangements doesn't depend on the particular vowel arrangement, right? So you can just multiply them.

Yeah, I have realized that by now. Thanks a lot for ur help :smile:
 

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