Answer

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Hint:- Find, total number of vowels and consonants in the given word.

As we know that in English alphabets A, E, I, O and U are vowels.

And the rest of the alphabets are consonants.

So, we can see that in the word DAUGHTER.

\[ \Rightarrow \]Total number of letters are 8. In which,

\[ \Rightarrow \]Total vowels \[ = {\text{ }}3\]

\[ \Rightarrow \]Total consonants \[ = 5\]

Now, we had to make a word using five letters of the given word.

Out of the five letters 2 should be vowels and 3 should be consonants.

\[ \Rightarrow \]So, here we had to choose 2 vowels out of the 3 vowels,

And that will be done in \[{}^3{C_2}\]\[{\text{ = 3 }}\]ways.

\[ \Rightarrow \]And 3 consonants out of 5 consonants.

And that will be done in \[{}^5{C_3}{\text{ = 10 }}\]ways.

\[ \Rightarrow \]And then arrange that five letters of word in \[5!\] ways.

So, total number of ways of forming a five letters word from the given word

DAUGHTER having 2 vowels and 3 consonants will be;

\[ \Rightarrow \]Total ways \[ = {}^3{C_2}*{}^5{C_3}*5! = 3600\].

Note:- Whenever we came up with this type of problems then first, we should

find the number of ways for selection of vowels and then find number of ways

for the selection of consonants. And at last never forget to multiply the number

of ways by \[n!\] to get the total number of ways. As it is given in the question that

a word of five letters can also be meaningless.

As we know that in English alphabets A, E, I, O and U are vowels.

And the rest of the alphabets are consonants.

So, we can see that in the word DAUGHTER.

\[ \Rightarrow \]Total number of letters are 8. In which,

\[ \Rightarrow \]Total vowels \[ = {\text{ }}3\]

\[ \Rightarrow \]Total consonants \[ = 5\]

Now, we had to make a word using five letters of the given word.

Out of the five letters 2 should be vowels and 3 should be consonants.

\[ \Rightarrow \]So, here we had to choose 2 vowels out of the 3 vowels,

And that will be done in \[{}^3{C_2}\]\[{\text{ = 3 }}\]ways.

\[ \Rightarrow \]And 3 consonants out of 5 consonants.

And that will be done in \[{}^5{C_3}{\text{ = 10 }}\]ways.

\[ \Rightarrow \]And then arrange that five letters of word in \[5!\] ways.

So, total number of ways of forming a five letters word from the given word

DAUGHTER having 2 vowels and 3 consonants will be;

\[ \Rightarrow \]Total ways \[ = {}^3{C_2}*{}^5{C_3}*5! = 3600\].

Note:- Whenever we came up with this type of problems then first, we should

find the number of ways for selection of vowels and then find number of ways

for the selection of consonants. And at last never forget to multiply the number

of ways by \[n!\] to get the total number of ways. As it is given in the question that

a word of five letters can also be meaningless.

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