Find the best tutors and institutes for Class 11 Tuition
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?
Solution :
Step 1: Analyis of the Given Word
The word "PERMUTATIONS" consists of $12$ letters. Let us identify the frequency of each letter:
- P: 1, E: 1, R: 1, M: 1, U: 1, T: 2, A: 1, I: 1, O: 1, N: 1, S: 1.
- Total letters ($n$) = 12.
- Repeating letters: 'T' appears 2 times.
Step 2: Defining the Constraint
We are required to arrange the letters such that there are exactly 4 letters between 'P' and 'S'.
Let the positions be represented by indices $1, 2, 3, \dots, 12$. If P is at position $i$, then S must be at position $i+5$ (to allow for 4 letters between them) or vice versa. The possible pairs of positions $(P, S)$ are:
$(1, 6), (2, 7), (3, 8), (4, 9), (5, 10), (6, 11), (7, 12)$.
There are $7$ such pairs. Since P and S can be swapped, there are $7 \times 2 = 14$ ways to place P and S.
Step 3: Calculating Permutations of Remaining Letters
Once P and S are fixed in a chosen pair of positions, there are $12 - 2 = 10$ remaining positions to be filled by the remaining $10$ letters.
The remaining letters are $\{E, R, M, U, T, T, A, I, O, N\}$.
The number of ways to arrange these 10 letters, where 'T' repeats 2 times, is given by the formula for permutations of multisets [Per Fundamental Principle of Counting]:
$$\text{Ways} = \frac{10!}{2!}$$
Calculating the value:
$$\frac{3,628,800}{2} = 1,814,400$$
Step 4: Final Computation
The total number of arrangements is the product of the number of ways to place P and S and the number of ways to arrange the remaining letters.
$$\text{Total Arrangements} = (\text{Ways to place P and S}) \times (\text{Ways to arrange remaining 10 letters})$$
$$\text{Total Arrangements} = 14 \times 1,814,400$$
$$\text{Total Arrangements} = 25,401,600$$
Visual Representation of Position Slots
Final Solution: The total number of ways to arrange the letters such that there are always 4 letters between P and S is 25,401,600.
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(ii): In how many ways can the letters of the word PERMUTATIONS be arranged if the (ii) vowels are all together,
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
Top Tutors who teach Permutations and Combinations
Top Tutor
Sector 54, Chandigarh
15 yrs of
Exp
11 students
353 hrs of classes completed
750
per hour
Online Classes
Highly qualified educator with over 15 years of experience specializing in Physics and Mathematics for grades 9–12. I bring 5+ years of dedicated classroom experience teaching classes 11 and 12, complemented by a decade of success in premier competitive coaching institutes. Expert in CBSE, ICSE, ISC, and IGCSE (Cambridge) curricula, with a proven track record of guiding students toward success in JEE Mains, NEET, NDA, and AFMC through practical, concept-driven instruction.
Suman Ma’am is an amazing physics teacher! She makes a very difficult subject feel easy and fun to learn. Her explanations are simple, and she is always incredibly patient when clearing our doubts. Thanks to her guidance, my confidence and marks in physics have improved so much. Highly recommended
Top Tutor
Vile Parle East, Mumbai
10 yrs of
Exp
5 students
201 hrs of classes completed
500
per hour
Online Classes
Tutor's home
Chemistry class 11 and cbse and state board free lance tutor online as well as teaching at my home
She was very good in explaining my 11th grade basics are strong because of her. She was very supportive and understanding.
Top Tutor
Katira Maharaja Hata, Arrah
4 yrs of
Exp
12 students
5807 hrs of classes completed
600
per hour
Online Classes
Tutor's home
I have successfully taught hundreds of Class 11 students across a diverse range of boards including CBSE, ICSE, IGCSE, International Curricula, and various State Boards. I specialize in teaching Physics and Mathematics for students from all streams—Science, Commerce, and Arts—ensuring tailored learning that suits individual academic needs. With extensive experience mentoring students for Class 10 to 12 , I provide focused training for JEE Foundation, JEE Main, and JEE Advanced aspirants, combining strong conceptual clarity with effective exam strategies to help students excel. Key Highlights: ✅ Expertise in major Indian and international syllabi ✅ Customized lesson plans catering to diverse learning styles ✅ Proven track record of consistent academic and competitive exam success ✅ Flexible one-on-one and group coaching options available Note: The fee listed here applies to personalized one-on-one sessions; group classes are offered at substantially discounted rates.
As a parent invested in my child's academic success, I've searched tirelessly for an online math tutor that not only imparts knowledge but also inspires confidence and fosters a genuine love for learning. Enter Shiwam - a beacon of excellence in online math education for Class 11th and IIT preparation.
Top Tutor
Peera Garhi, Delhi
8 yrs of
Exp
2 students
72 hrs of classes completed
700
per hour
Online Classes
Student's Home
Tutor's home
I am having more than 8+ years of experience to teach the Computer Science/ informatic Practice subject to the students of 11th class. I can teach all boards students.
Top Tutor
A B Road, Indore
4 yrs of
Exp
1 student
Online Classes
Student's Home
I am a professional tutor with four years of experience in teaching. Having a degree in M.Com from Vikram University and an MBA from IMD Indore I have also completed my B.Ed from IPS Academy Indore in the year 2015. I teach Spoken English, Accountancy, Business Studies, Economics, Statistics, Secretarial Practices, etc., for State, CBSE, ISC/ICSE Board students. I am well versed in English, Hindi and can take classes at Student's Home, My Home and Online Classes via video chat, skype, google hangout, etc.
Other Subjects in CBSE - Class 11
Download free CBSE - Class 11 Mathematics Permutations and Combinations EXERCISE 6.3 worksheets
Download Now
