Finding the Number of Words with Fixed Vowel Order

In summary, the conversation discusses the process of finding the number of different 11-letter words formed from the letters MTHMTCS, with the restriction that the vowels AEAI must occur in the same order. The solution involves finding the number of vowel arrangements and multiplying it by the number of ways to place the remaining consonants.
  • #1
ritwik06
580
0

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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  • #2
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??
 
  • #3
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!
 
  • #4
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 betwen each vowel doesn't help. I have also tried to fix the consonants first. Help :smile:
regards
 
Last edited:
  • #5
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.
 
  • #6
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:
 

1. What is the difference between permutation and combination?

Permutation and combination are both methods of counting the number of ways to arrange a set of objects. However, the main difference is that permutation takes into account the order of the objects, while combination does not. In other words, permutation considers the arrangement of objects as important, while combination considers only the selection of objects.

2. How do I calculate the number of permutations?

The formula for calculating the number of permutations is n! / (n-r)!, where n represents the total number of objects and r represents the number of objects being selected. This formula is often written as nPr, where n is the total number of objects and r is the number of objects being selected.

3. What is the difference between a permutation with repetition and a permutation without repetition?

A permutation with repetition occurs when there are repeated objects in the set being arranged. This means that some of the objects are identical and the order in which they are arranged does not matter. On the other hand, a permutation without repetition occurs when all the objects in the set are unique and the order in which they are arranged does matter.

4. How do I calculate the number of combinations?

The formula for calculating the number of combinations is n! / (r!(n-r)!), where n represents the total number of objects and r represents the number of objects being selected. This formula is often written as nCr, where n is the total number of objects and r is the number of objects being selected.

5. In what real-life situations are permutation and combination used?

Permutation and combination are used in various fields such as mathematics, statistics, computer science, and engineering. In real life, these concepts are applied in areas such as genetics, probability, and cryptography. For example, in genetics, permutation and combination are used to study gene sequences and the probability of certain traits being passed on to offspring. In cryptography, permutation and combination are used to create complex codes and passwords.

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