How many words can be formed the letters of the word father so that the vowels always come together?

$\begingroup$

Arrange the letters(every arrangement must contain all letters of the word) of the word 'BENGALI', so that no two vowels are together.

What my cute little brain could find out:
Let me first arrange the vowels...
__ E __ A __ I__
where the underscores contain the consonants. Now, clearly, there will be $^{3}P_3$ arrangements. So my brain tells me to find the ans for the E A I one and then multiply it by $^{3}P_3$
Now my brain thinks for a minute and then says:
"Hey! There are $4$ underscores and how many consonants do you have? Its $4$ Is it not a modified stars and bars problem?"
I thought for a moment, and agreed with my brain. Then it said:
"Find all integer solutions to the equation based on the following conditions:
$x_1+x_2+x_3+x_4=4$, where $x_1,x_4≥0$ and $x_2,x_3≥1$"
And the answer to this is $\binom{2 + 4 - 1}{2} = \binom{2 + 4 - 1}{4 - 1}$(just some honesty!)
"But wait! There are $^{4}P_4$ ways of arranging the consonants. So multiply this by $^{4}P_4$"
And finally by $^{3}P_3$
So my final answer is
$$\binom{2 + 4 - 1}{2}\times^{4}P_4 \times^{3}P_3$$ Am I correct? If yes, is there any better or more efficient way? If yes, would you show that?

asked Dec 10, 2017 at 17:18

ami_baami_ba

2,0523 gold badges11 silver badges31 bronze badges

$\endgroup$

9

$\begingroup$

There are $4$ consonants, hence $5$ slots to place one of the three vowels. The consonants as well as the vowels can be written in any order. It follows that there are $${5\choose3}\cdot 3!\cdot 4!=1440$$ admissible arrangements of the $7$ letters.

answered Dec 10, 2017 at 18:37

Christian BlatterChristian Blatter

219k13 gold badges172 silver badges437 bronze badges

$\endgroup$

1

Disclaimer

The questions posted on the site are solely user generated, Doubtnut has no ownership or control over the nature and content of those questions. Doubtnut is not responsible for any discrepancies concerning the duplicity of content over those questions.

In how many different ways, can the letters of the words EXTRA be arranged so that the vowels are never together?

1. 168
2. 48
3. 120
4. 72

Answer (Detailed Solution Below)

Option 4 : 72

Calculation:

EXTRA → Total number of words = 5 and total number of vowels = 2

The word EXTRA can be arranged in 5! ways = 120 ways

The word EXTRA can be arranged in such a way that the vowels will be together = 4! × 2!

⇒ (4 × 3 × 2 × 1) × (2 × 1)

⇒ 48 ways

The letters of the words EXTRA be arranged so that the vowels are never together = (120 - 48) = 72 ways.

∴ The letters of the words EXTRA be arranged so that the vowels are never together in 72 ways.

How many words can be formed from the letters of the word ‘DAUGHTER’ so that(i) The vowels always come together?(ii) The vowels never come together?

Answer

Verified

Hint: The word daughter has $8$ letters in which $3$ are vowels. For the vowels to always come together consider all the $3$ vowels to be one letter (suppose V) then total letters become $6$ which can be arranged in $6!$ ways and the vowels themselves in $3!$ ways.Complete step-by-step answer:
Given word ‘DAUGHTER’ has $8$ letters in which $3$ are vowels and 5 are consonants. A, U, E are vowels and D, G, H, T, R are consonants.
(i)We have to find the total number of words formed when the vowels always come together.
Consider the three vowels A, U, E to be one letter V then total letters are D, G, H, T, R and V. So the number of letters becomes $6$
So we can arrange these $6$ letters in $6!$ ways. Since the letter V consists of three vowels, the vowels themselves can interchange with themselves. So the number of ways the $3$vowels can be arranged is $3!$
Then,
$ \Rightarrow $ The total number of words formed will be=number of ways the $6$ letters can be arranged ×number of ways the $3$ vowels can be arranged
On putting the given values we get,
$ \Rightarrow $ The total number of words formed=$6! \times 3!$
We know $n! = n \times \left( {n - 1} \right)! \times ...3,2,1$
$ \Rightarrow $ The total number of words formed=$6 \times 4 \times 5 \times 3 \times 2 \times 1 \times 3 \times 2 \times 1$
On multiplying all the numbers we get,
 $ \Rightarrow $ The total number of words formed=$24 \times 5 \times 6 \times 6$
$ \Rightarrow $ The total number of words formed=$120 \times 36$
$ \Rightarrow $ The total number of words formed=$4320$
The number of words formed from ‘DAUGHTER’ such that all vowels are together is $4320$.

(ii)We have to find the number of words formed when no vowels are together.
Consider the following arrangement- _D_H_G_T_R
The spaces before the consonants are for the vowels so that no vowels come together. Since there are $5$ consonants so they can be arranged in $5!$ ways.
There are $6$ spaces given for $3$ vowels. We know to select r things out of n things we write use the following formula-${}^{\text{n}}{{\text{C}}_{\text{r}}}$=$\dfrac{{n!}}{{r!n - r!}}$
So to select $3$ spaces of out $6$ spaces =${}^6{{\text{C}}_3}$
And the three vowels can be arranged in these three spaces in $3!$ ways.
$ \Rightarrow $ The total number of words formed=${}^6{{\text{C}}_3} \times 3! \times 5!$
$ \Rightarrow $ The total number of words formed=$\dfrac{{6!}}{{3!6 - 3!}} \times 5! \times 3!$
$ \Rightarrow $ The total number of words formed=$\dfrac{{6!}}{{3!}} \times 5!$
On simplifying we get-
$ \Rightarrow $ The total number of words formed=$\dfrac{{6 \times 5 \times 4 \times 3!}}{{3!}} \times 5!$
$ \Rightarrow $ The total number of words formed=$120 \times 5 \times 4 \times 3 \times 2 \times 1$
On multiplying we get,
$ \Rightarrow $ The total number of words formed=$14400$
The total number of words formed from ‘DAUGHTER’ such that no vowels are together is $14400$.

Note: Combination is used when things are to be arranged but not necessarily in order. Permutation is a little different. In permutation, order is important. Permutation is given by-
$ \Rightarrow {}^n{P_r} = \dfrac{{n!}}{{n - r!}}$ Where n=total number of things and r=no. of things to be selected.

How many ways are there to arrange the letters of the word father with vowels?

How many words can be formed when vowels are always together?

How many words can be formed using all the letters of the word father?

How many words can be formed in letters of i so the vowels always come together II the vowels never come together?