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How many words can we make for the PERMUTATIONS word using only one vo

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Bunuel
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How many words can we make for the PERMUTATIONS word using only one vo [#permalink] New post   29 Dec 2021, 08:45
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How many words can we make for the PERMUTATIONS word using only one vowel, and 2 consonant that the vowel always been in middle?

A) 154
B) 155
C) 156
D) 157
E) 158


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gmatophobia
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Re: How many words can we make for the PERMUTATIONS word using only one vo [#permalink] New post   29 Dec 2021, 23:11
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Let's break PERMUTATIONS into its vowels and consonants -

Vowels - {P,R,M,TT,N,S}

Consonants - {E,U,I,O,A}

So we have six different vowels and 5 different consonants.

Also, we have two consonant which are similar.

Case 1 : The two consonants selected are different

Number of ways of choosing a vowel (middle alphabet of the word) = 5 C 1 = 5

Number of ways of choosing a consonant = 6 P 2 = 30

Total ways = 30 * 5 = 150 ways

Case 2 : The two consonants selected are same

Number of ways of choosing a vowel (middle alphabet of the word) = 5 C 1 = 5

Number of ways of choosing a consonant = 1 (Two Ts)

Total ways = 1 * 5 = 5 ways

Total ways = 150 + 5 = 155 ways

IMO - B
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Re: How many words can we make for the PERMUTATIONS word using only one vo [#permalink] New post   16 Jan 2022, 11:11
Does "that the vowel always been in middle?" mean anything? I guess it doesn't really matter to solve the question
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Re: How many words can we make for the PERMUTATIONS word using only one vo [#permalink] New post   16 Jan 2022, 12:01
Can anyone help me with the answer Explantation please?

Also please correct me for the below steps :

Total Vowels : 5 (EUAIO)
Total Consonants: 7 (Remaining letters)

3 places to be filled : — — —
2 consonants can be chosen in 7C2 ways
2 consonants can be arranged in 2! Ways
Middle place to be filled with vowel , hence, 5C1 ways

Hence total ways : 7C2 * 2! * 5C1 = 210.

Can you please highlight where am I going wrong here and what am I missing out???
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Re: How many words can we make for the PERMUTATIONS word using only one vo [#permalink] New post   16 Jan 2022, 12:27
Anamika1496 wrote:
Can anyone help me with the answer Explantation please?

Also please correct me for the below steps :

Total Vowels : 5 (EUAIO)
Total Consonants: 7 (Remaining letters)

3 places to be filled : — — —
2 consonants can be chosen in 7C2 ways
2 consonants can be arranged in 2! Ways
Middle place to be filled with vowel , hence, 5C1 ways

Hence total ways : 7C2 * 2! * 5C1 = 210.

Can you please highlight where am I going wrong here and what am I missing out???


Hi,

There are two T in the word and they will form the same set of words, hence you are double counting the words in 7C2.

I think you need to consider cases and solve this. I have shared my attempt of the question, let me know if you have questions.
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Re: How many words can we make for the PERMUTATIONS word using only one vo [#permalink] New post   23 Apr 2024, 12:22
I tried this way;
no of ways of selection of vowels =5
whatever will be selection procedure for consonants ,product of both will be a multiple of 5.
Hence 155.
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Re: How many words can we make for the PERMUTATIONS word using only one vo [#permalink]
23 Apr 2024, 12:22
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