Permutations and Combinations Class 11 Notes

• Permutations with repetition allowed: each of the r positions has n choices, giving n^r arrangements.
• All n objects arranged in a row: n! permutations.
• Arranging n objects in a circle: (n − 1)! circular permutations (fixing one object eliminates rotational duplicates).

When n objects include p of one kind, q of another, r of a third, and so on, the number of distinct arrangements is $\frac{n!}{p!q!r!\cdots }$. For example, the letters of MISSISSIPPI (M=1, I=4, S=4, P=2) give $\frac{11!}{1!4!4!2!}=34650$ distinct arrangements.

• If clockwise and anticlockwise order are both the same (e.g., a necklace), the count is (n − 1)!/2.
• If direction matters (e.g., people seated around a table with a distinguished side), the count is (n − 1)!.
• Fixing one object is the standard technique — it removes all rotational duplicates.

• Relation between permutations and combinations:
• ${}^n{P}_r=r!{\times }^n{C}_r$
• .
• Symmetry property:
• ${}^n{C}_r{=}^n{C}_{n-r}$
• .
• Pascal's identity:
• ${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$
• .
• Sum of all combinations:
• ${\sum }_{r=0}^n{}^n{C}_r=2^n$
• .

• nPr = r! · nCr — permutations count order, combinations do not.
• nCr = nC(n − r) — choosing r to include equals choosing n − r to exclude.
• nCr + nC(r − 1) = (n + 1)Cr — Pascal's recurrence, the basis of Pascal's triangle.
• For r objects taken one at a time with repetition from n objects: n^r ways.
• When all n objects are taken (r = n): nPn = nCn = 1 arrangement/selection.

Example: In how many ways can a committee of 4 men and 3 women be chosen from 7 men and 5 women?

Solution: Choose 4 men from 7: ${}^7{C}_4=35$. Choose 3 women from 5: ${}^5{C}_3=10$. By the multiplication principle, total = 35 × 10 = 350 ways.

Example: Find the number of permutations of the letters of the word ARRANGE.

Solution: ARRANGE has 7 letters with A repeated 2 times and R repeated 2 times. Distinct arrangements = $\frac{7!}{2!2!}=\frac{5040}{4}=1260$.

Exercises (NCERT Q1–Q31)

• 1) How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed? (ii) repetition is not allowed?
• 2) How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
• 3) How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
• 4) How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
• 5) A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
• 6) Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
• 7) Evaluate: (i) 8! (ii) 4! − 3!.
• 8) Is 3! + 4! = 7! ?
• 9) Compute 8! / (6! × 2!).
• 10) If 1/6! + 1/7! = x/8!, find x.
• 11) Evaluate n!/(n − r)! when (i) n = 6, r = 2; (ii) n = 9, r = 5.
• 12) How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
• 13) How many 4-digit numbers are there with no digit repeated?
• 14) How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?
• 15) 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?
• 16) From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position?
• 17) Find n if ⁿ⁻¹P₃ : ⁿP₄ = 1 : 9.
• 18) Find r if (i) ⁵Pᵣ = 2 · ⁶Pᵣ₋₁, (ii) ⁵Pᵣ = ⁶Pᵣ₋₁.
• 19) How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?
• 20) 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, (ii) all letters are used at a time, (iii) all letters are used but the first letter is a vowel?
• 21) In how many of the distinct permutations of the letters in MISSISSIPPI do the four I's not come together?
• 22) In how many ways can the letters of the word PERMUTATIONS be arranged if (i) words start with P and end with S, (ii) vowels are all together, (iii) there are always 4 letters between P and S?
• 23) If ⁿC₈ = ⁿC₂, find ⁿC₂.
• 24) Determine n if (i) ²ⁿC₃ : ⁿC₃ = 12 : 1, (ii) ²ⁿC₃ : ⁿC₃ = 11 : 1.
• 25) How many chords can be drawn through 21 points on a circle?
• 26) In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
• 27) Find the number of ways of selecting 9 balls from 6 red, 5 white and 5 blue balls if each selection consists of 3 balls of each colour.
• 28) Determine the number of 5-card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
• 29) In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl, if each cricket team of 11 must include exactly 4 bowlers?
• 30) A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
• 31) In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

Answers at a glance

1. 1
   Q1 → (i) 125, (ii) 60.
2. 2
   Q2 → 108.
3. 3
   Q3 → 5040.
4. 4
   Q4 → 336.
5. 5
   Q5 → 8.
6. 6
   Q6 → 20.
7. 7
   Q7 → (i) 40320, (ii) 18.
8. 8
   Q8 → No (30 ≠ 5040).
9. 9
   Q9 → 28.
10. 10
    Q10 → x = 64.
11. 11
    Q11 → (i) 30, (ii) 15120.
12. 12
    Q12 → 504.
13. 13
    Q13 → 4536.
14. 14
    Q14 → 60.
15. 15
    Q15 → 120 total, 48 even.
16. 16
    Q16 → 56.
17. 17
    Q17 → n = 9.
18. 18
    Q18 → (i) r = 3, (ii) r = 4.
19. 19
    Q19 → 40320.
20. 20
    Q20 → (i) 360, (ii) 720, (iii) 240.
21. 21
    Q21 → 33810.
22. 22
    Q22 → (i) 1814400, (ii) 2419200, (iii) 25401600.
23. 23
    Q23 → 45.
24. 24
    Q24 → (i) n = 5, (ii) n = 6.
25. 25
    Q25 → 210.
26. 26
    Q26 → 40.
27. 27
    Q27 → 2000.
28. 28
    Q28 → 778320.
29. 29
    Q29 → 3960.
30. 30
    Q30 → 200.
31. 31
    Q31 → 35.

MCQ Questions

1 Mark Questions

1. 1
   Write the value of 0!. (Ans: 1)
2. 2
   State the formula for nPr. (Ans: n!/(n − r)!)
3. 3
   Find C(4, 1). (Ans: 4)
4. 4
   In how many ways can a committee of 2 be chosen from 5 people? (Ans: C(5, 2) = 10)

2 Mark Questions

1. 1
   How many 3-digit numbers can be formed from the digits 1–5 without repetition? (Ans: 5 × 4 × 3 = 60)
2. 2
   How many arrangements of the word 'APPLE' are there? (Ans: 5!/2! = 60)
3. 3
   Find the number of diagonals of a hexagon using the combination formula. (Ans: C(6, 2) − 6 = 9)

3 Mark Questions

1. 1
   If nP3 = 60, find n. (Ans: n!/(n−3)! = 60 ⇒ n = 5)
2. 2
   A team of 3 boys and 2 girls is to be selected from 5 boys and 4 girls; in how many ways? (Ans: C(5,3) × C(4,2) = 10 × 6 = 60)
3. 3
   How many 4-letter words can be formed from the letters of the word 'EXAMINATION' without repetition, taking vowels always together? (Ans: treat the 5 vowels {E,A,A,I,O} converted to distinct with repeats — count via cases)

4 Mark Questions

1. 1
   In how many ways can the letters of the word 'INDEPENDENCE' be arranged (i) without restriction, (ii) when the vowels always come together, (iii) when the word starts with I and ends with E? (Ans: (i) 12!/(3!2!4!), (ii) vowel block count, (iii) fixed ends) Show the factorial divisions.
2. 2
   Prove the identity C(n, r) = C(n, n − r) using the factorial definition, and verify with n = 10, r = 3. Then find the number of ways to form a hand of 3 cards from a 52-card pack. (Ans: C(52, 3))