Find the best tutors and institutes for Class 11 Tuition
Q11(ii):
In how many ways can the letters of the word PERMUTATIONS be arranged if the (ii) vowels are all together,
Solution :
Step 1: Analyzing the Word Composition
The given word is PERMUTATIONS. We perform an exhaustive count of its constituent letters:
- Total letters ($n$) = 12
- Letter frequencies:
- P: 1, E: 1, R: 1, M: 1, U: 1, T: 2, A: 1, I: 1, O: 1, N: 1, S: 1
- Total distinct letters: 11 distinct types, with 'T' repeating twice.
Step 2: Segregating Vowels and Consonants
To arrange the letters such that all vowels are kept together, we identify the specific vowels in the word:
- Vowels: {E, U, A, I, O} (Total = 5)
- Consonants: {P, R, M, T, T, N, S} (Total = 7)
[By the method of grouping, we treat the set of vowels as a single entity or 'super-letter' (V)].
Step 3: Calculating Internal Permutations of the Vowel Group
The 5 vowels {E, U, A, I, O} are all distinct. The number of ways to arrange these 5 vowels internally is given by $5!$ [Per the Multiplication Principle of counting]:
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
Step 4: Arranging the Combined Entities
After grouping the vowels into a single unit (V), we now have the following entities to arrange:
- Consonants: {P, R, M, T, T, N, S} (7 entities)
- Vowel group: {V} (1 entity)
- Total entities to arrange = $7 + 1 = 8$
Among these 8 entities, the letter 'T' appears twice. Therefore, the number of distinct permutations of these 8 entities is calculated as:
$\frac{8!}{2!}$
Calculation: $\frac{40320}{2} = 20160$
Step 5: Final Combinatorial Synthesis
The total number of arrangements is the product of the external arrangements of the entities and the internal arrangements of the vowels [Per the Fundamental Counting Principle]:
$\text{Total} = (\text{Arrangements of entities}) \times (\text{Internal vowel permutations})$
$\text{Total} = 20160 \times 120$
$\text{Total} = 2,419,200$
Visual Representation of the Grouping Method
Final Solution: 2,419,200 ways
More Questions from Class 11 Mathematics Permutations and Combinations EXERCISE 6.3
- Q1: How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
- Q2: How many 4-digit numbers are there with no digit repeated?
- Q3: How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?
- Q4: Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?
- Q5: From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person can not hold more than one position?
- Q6: Find $n$ if ${}^{n–1}P_3 : {}^nP_4 = 1 : 9$.
- Q7(i): Find $r$ if (i) ${}^5P_r = 2 \cdot {}^6P_{r-1}$
- Q7(ii): Find $r$ if (ii) ${}^5P_r = {}^6P_{r-1}$.
- Q8: How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?
- Q9(i): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (i) 4 letters are used at a time,
- Q9(ii): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (ii) all letters are used at a time,
- Q9(iii): How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if. (iii) all letters are used but first letter is a vowel?
- Q10: In how many of the distinct permutations of the letters in MISSISSIPPI do the four $I$'s not come together?
- Q11(i): In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) words start with P and end with S,
- Q11(iii): In how many ways can the letters of the word PERMUTATIONS be arranged if the (iii) there are always 4 letters between P and S?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
Top Tutors who teach Permutations and Combinations
Top Tutor
Malviya Nagar, Delhi
15 yrs of
Exp
7 students
3 hrs of classes completed
400
per hour
Student's Home
I have 15 years experience in home tuition
I have taken tutions from Ramesh Sir and he always made it sure to clarify all my account related doubts. Best part is he is available 24*7 to solve all the queries and he is one of the best faculty in Delhi for accounts.
Top Tutor
Indirapuram, Ghaziabad
5 yrs of
Exp
39 hrs of classes completed
Online Classes
I was a working professional .Left my job and started teaching as a tutor. I have been a national level debater . I am a famous teacher amongst my students .
Top Tutor
Tilak Nagar, Delhi
8 yrs of
Exp
9 students
486 hrs of classes completed
800
per hour
Online Classes
I can teach the student from the very basic level to up to Jee advance and neet level. I can clear all the doubt regarding organic, physical and inorganic.
An exceptional educator who transforms a notoriously challenging subject into something genuinely engaging. They have a rare gift for breaking down complex concepts—like abstract thermodynamic laws or intricate reaction mechanisms—into clear, relatable explanations. Lessons are always interactive and well-paced, blending theory with memorable practical examples that make formulas stick. Beyond their deep subject knowledge, they are incredibly approachable, infinitely patient during office hours, and truly invested in student success. They don’t just teach chemistry for an exam; they foster a genuine curiosity about how the world works at a molecular level. Highly recommended!
Top Tutor
Bade Kajipur, Gorakhpur
20 yrs of
Exp
10 students
1221 hrs of classes completed
350
per hour
Online Classes
Student's Home
Tutor's home
This is 12 Month NEET Physics Syllabus Crash Course Starting From 1 April 2026 to 30 April 2027. Four Days in a Week. Every Live Class is of 1hour 30 minutes, including Precise Notes of each topic Practice Questions and Online Test With Video Solutions. Formula Sheet for every chapter. Course Fees of NEET 2026 Crash Course for Physics is 9000/- I prefer to teach physics specially class 11 and 12, I do focus on Competitive level Physics like N.E.E.T. and I.I.T.J.E.E. along with basic physics to make my students understand physics easier. My focus is on competitive exam for betterment of students. This is 12 Month NEET Physics Syllabus Crash Course Starting From 1 April 2026 to 30 April 2027. Four Days in a Week. I have been teahing Physics twenty years of teaching experience, I had done my schooling and college from Gorakhpur city Uttar Pradesh. I am Teaching since 2004 in Gorakhpur initially off-line, and teaching online since 2020 with zoom and Google Meet I do provide study material like Notes, Question Bank and D.P.P., No Additional charge is require to pay, it is included in class fees. Monthly Fees 4000/- for 3 days in a week and in a month 12 hours
Your positive attitude and enthusiasm for teaching have made a lasting impression on me. Thank you for being a teacher who truly cares about their students’ success and well-being.
Top Tutor
Suddagunte Palya Balaji Nagar, Bangalore
Less than a yr
Exp
2 students
27 hrs of classes completed
1000
per hour
Online Classes
I am a Ph.D. Scholar. I completed my BCA from Calicut University, finished my MCA from Kristu Jayanti College (Autonomous), and currently pursuing my Ph.D. in Christ (Deemed to be University). Now, I am doing my research in Artificial Intelligence (Computer Science) and Exoplanetary Science (Physics). Therefore, I have relevant knowledge and experience in Computer Science and Physics concepts, including advanced topics like AI, ML, DL, NLP, Quantum Mechanics, Astrophysics, etc...
Other Subjects in CBSE - Class 11
Download free CBSE - Class 11 Mathematics Permutations and Combinations EXERCISE 6.3 worksheets
Download Now
