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Solutions

Overview

Step-by-step NCERT solutions for Permutations and Combinations (Chapter 6, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Permutations and Combinations textbook chapter.

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What these solutions cover

All 42 questions in Permutations and Combinations are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 6.1

  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 of the digits 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?

Exercise 6.2

  1. Evaluate
    • (i) 8!
    • (ii) 4! – 3!
  2. Is 3! + 4! = 7! ?
  3. Compute 8! / (6! × 2!)
  4. If 1/6! + 1/7! = x/8!, find x.
  5. Evaluate n! / (n – r)!, when
    • (i) n = 6, r = 2
    • (ii) n = 9, r = 5.

Exercise 6.3

  1. How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
  2. How many 4-digit numbers are there with no digit repeated?
  3. How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?
  4. 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?
  5. 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?
  6. Find n if (n – 1)P₃ : nP₄ = 1 : 9.
  7. Find r if
    • (i) ⁵Pᵣ = 2 · ⁶Pᵣ₋₁
    • (ii) ⁵Pᵣ = ⁶Pᵣ₋₁
  8. How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once?
  9. 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 first letter is a vowel?
  10. In how many of the distinct permutations of the letters in MISSISSIPPI do the four I's not come together?
  11. In how many ways can the letters of the word PERMUTATIONS be arranged if the
    • (i) words start with P and end with S,
    • (ii) vowels are all together,
    • (iii) there are always 4 letters between P and S?

Exercise 6.4

  1. If nC₈ = nC₂, find nC₂.
  2. Determine n if
    • (i) ²ⁿC₃ : nC₃ = 12 : 1
    • (ii) ²ⁿC₃ : nC₃ = 11 : 1
  3. How many chords can be drawn through 21 points on a circle?
  4. In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?
  5. Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.
  6. Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
  7. 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?
  8. 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.
  9. 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?

Miscellaneous Exercise

  1. How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER?
  2. How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?
  3. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of:
    • (i) exactly 3 girls?
    • (ii) at least 3 girls?
    • (iii) at most 3 girls?
  4. If the different permutations of all the letters of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?
  5. How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated?
  6. The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?
  7. In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?
  8. Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.
  9. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?
  10. From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?
  11. In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?
Sample

Worked examples

3 questions from Permutations and Combinations, solved step by step.

Exercise 6.1

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 of the digits is not allowed?

(i) With repetition allowed:

Each of the 3 places can be filled by any of the 5 digits.

By the multiplication principle, the number of 3-digit numbers = 5 × 5 × 5 = 125.

(ii) Without repetition:

The hundreds place can be filled in 5 ways. After choosing one digit, the tens place can be filled in 4 ways, and the units place in 3 ways.

By the multiplication principle, the number of 3-digit numbers = 5 × 4 × 3 = 60.

Answer: (i) 125, (ii) 60

Exercise 6.2

Evaluate n! / (n – r)!, when (i) n = 6, r = 2 (ii) n = 9, r = 5.

(i) n = 6, r = 2:

n! / (n – r)! = 6! / (6 – 2)! = 6! / 4! = (6 × 5 × 4!) / 4! = 6 × 5 = 30.

(ii) n = 9, r = 5:

n! / (n – r)! = 9! / (9 – 5)! = 9! / 4! = 9 × 8 × 7 × 6 × 5 = 15120.

Answer: (i) 30, (ii) 15120

Exercise 6.3

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

We need to arrange 3 digits from 9 distinct digits (1–9) with no repetition.

This is the number of permutations of 9 objects taken 3 at a time:

⁹P₃ = 9! / (9 – 3)! = 9! / 6! = 9 × 8 × 7 = 504.

Answer: 504

All 42 questions in Permutations and Combinations are solved step by step in the Solutions PDF above.

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