Answer
417.9k+ views
Hint: Here, the given question is from permutation and combination. We have to find the number of arrangements that can be formed out by the letters of the word “LOGARITHM” considering different conditions. So, firstly we have to count the number of consonants and vowels in the word “LOGARITHM”, then apply the conditions such as in the first question arrange the consonant by putting a space between two consonants and then arrange the vowels in that place.
Complete step-by-step solution:
(1) Here, the letters of the word “LOGARITHM” are arranged in such a way that no two vowels come together.
Since, the given condition is that no two vowels come together and total six consonants and three vowels are in the word “LOGARITHM” so the possible arrangement may be VCVCVCVCVCVCV. And of the six positions of consonant, six consonants can be arranged in $6!$ ways. The arrangement of six consonants produces seven places for vowels and we have only three vowels so this can be arranged as $7{p_3}$.
So, the total possible number of arrangements is $6!\,7{p_3}$.
Hence, option (A) is correct.
(2) Here, the letters of the word “LOGARITHM” are arranged in such a way that all the vowels do not come together.
To arrange according to given condition, find the total possible number of arrangements of letters without any condition that is $9!$ and then subtract the case when all the vowels come together from this.
Now, put all vowels in to a box and consider it as a single letter and make arrangements of six consonants and a box of vowels that comes as $7!\,3!$ because the seven letters can be arranged in $7!$ and the vowels inside the box can be arranged internally in $3!$ ways.
So, the total possible number of arrangements is $9! - 7!\,3!$
Hence, option (B) is correct.
(3) Here, the letters of the word “LOGARITHM” are arranged in such a way that no two consonants come together.
We have only three vowels but six consonants and we have to arrange such that no two consonants come together and this can be possible if we arrange the letters as CVCVCVC but there is only four places for consonants but we have to arrange six consonants so, it is not possible to arrange in such a way.
So, the possible number of arrangements is zero.
Hence, option (A) is correct.
Note: We should know,
$n{p_r} = \dfrac{{n!}}{{\left( {n - r} \right)!\,}}$
\[n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!\,r!}}\]
It will be helpful in finding the numerical value.
Complete step-by-step solution:
(1) Here, the letters of the word “LOGARITHM” are arranged in such a way that no two vowels come together.
Since, the given condition is that no two vowels come together and total six consonants and three vowels are in the word “LOGARITHM” so the possible arrangement may be VCVCVCVCVCVCV. And of the six positions of consonant, six consonants can be arranged in $6!$ ways. The arrangement of six consonants produces seven places for vowels and we have only three vowels so this can be arranged as $7{p_3}$.
So, the total possible number of arrangements is $6!\,7{p_3}$.
Hence, option (A) is correct.
(2) Here, the letters of the word “LOGARITHM” are arranged in such a way that all the vowels do not come together.
To arrange according to given condition, find the total possible number of arrangements of letters without any condition that is $9!$ and then subtract the case when all the vowels come together from this.
Now, put all vowels in to a box and consider it as a single letter and make arrangements of six consonants and a box of vowels that comes as $7!\,3!$ because the seven letters can be arranged in $7!$ and the vowels inside the box can be arranged internally in $3!$ ways.
So, the total possible number of arrangements is $9! - 7!\,3!$
Hence, option (B) is correct.
(3) Here, the letters of the word “LOGARITHM” are arranged in such a way that no two consonants come together.
We have only three vowels but six consonants and we have to arrange such that no two consonants come together and this can be possible if we arrange the letters as CVCVCVC but there is only four places for consonants but we have to arrange six consonants so, it is not possible to arrange in such a way.
So, the possible number of arrangements is zero.
Hence, option (A) is correct.
Note: We should know,
$n{p_r} = \dfrac{{n!}}{{\left( {n - r} \right)!\,}}$
\[n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!\,r!}}\]
It will be helpful in finding the numerical value.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)