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CBSE Class 11 · Mathematics

Permutations and Combinations

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Permutations and Combinations” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 11
  • Medium level
  • 8 concepts

1Fundamental Principle of Counting

The fundamental counting principle: if one job has m ways and a next independent job has n ways, together there are m × n ways. 3 shirts and 4 trousers give 12 outfits. Add when the jobs are alternatives that cannot both run; multiply when they run in sequence as taught.

Adding 3+4 for outfits is the miss.

Figure. Two independent choices multiply: 2 shirts and then 3 trousers give 6 outfits. Count the leaves, do not add 2 + 3.

How it works

  1. Name the jobs and whether they both runSequence or alternative.
  2. Multiply for a sequence, add for an exclusive or — as taughtThe principle.
  3. 12 from 3×4, not 7The count.

3 shirts, 4 trousers

How many shirt-and-trouser outfits?

  • Jobsshirt then trouser — both run
  • Ways3 × 4
  • Outfits12

Pro tip. Multiply when both jobs run.

2Permutations

A permutation is an arrangement — order matters. P(n, r) = n! / (n−r)! as taught: 5 people in 3 chairs is 5×4×3 = 60. ABC and ACB are two permutations. A leftover unused person is not in that arrangement.

Treating ABC and ACB as one because they use the same letters is the combination-steal.

Figure. A permutation fills ordered places. Three distinct people in three seats: 3 choices, then 2, then 1, so 3P3 = 3! = 6 arrangements.

How it works

  1. Ask whether order mattersYes → permute.
  2. Write n(n−1)… down r factors, or the factorial formP(n,r).
  3. Keep ABC ≠ ACBArrangement.

3Combinations

A combination is a team — order does not matter. C(n, r) = n! / (r!(n−r)!) as taught: 5 people, choose 3, is 10. ABC and ACB are one combination. Combination is the subset-count; permutation is those subsets arranged.

Using 60 as the team-count for 5 choose 3 is the arrange-steal.

Figure. A combination is a team, not a queue. From {A, B, C} the two-person teams are only three: AB, AC, BC. Swapping two names does not make a new team.

How it works

  1. Ask whether order mattersNo → combine.
  2. Write C(n,r)The team-count.
  3. Note C(5,3)=10 = P(5,3)/3!Teams, then arrangements of a team.

4A definition is a test you can run

Permute / combine is a test: order-matters or not, plus the factorial write. If you only say “ways”, you have a heading.

A pretty 5! badge is not the test.

Figure. Run the order test before you pick a formula. If swapping two chosen names changes the outcome, it is a permutation; if the same set is the same outcome, it is a combination.

How it works

  1. Name n, r, and whether order mattersThe object.
  2. Write P or CThe test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given is n, r, and “arrange or team”. The unknown is the count. Copy “order matters” before you pick C.

Using n=3 because you saw 3 chairs, when 5 people were sitting down, is a silent given.

Figure. Write n (how many are available) and r (how many places) before any formula. Five people and two ordered seats is 5P2 = 5 × 4 = 20, not a bare 20.

How it works

  1. Copy n, r, and the order-wordThe given.
  2. Name P or CThe unknown.
  3. Then the factorial writeGiven first.

6One worked case is enough at this class

One count is enough. A page of the same 12-outfit clone does not add a new C(5,3).

Ten twins of 60 are still one idea.

Figure. One school case is enough: 5 distinct people, 2 ordered seats. First seat 5 ways, second seat 4 ways, product 20. That is 5P2.

How it works

  1. Run one multiply or one P or CThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A locker-photo is a setting. The maths is m×n or P or C. A padlock is not 5!.

A team-photo still needs “order or not”.

Figure. A committee does not sit in numbered chairs. 5P2 = 20 counts ordered pairs; each team is counted twice (AB and BA), so the school answer is 5C2 = 10.

How it works

  1. Keep the locker as a storyA setting.
  2. Write the jobs or n, rThe maths.
  3. Then multiply, P, or CDo not swap.

8Check by the opposite action or the opposite test

Check P by writing the falling product; check C by P/r!. 60/6=10 for 5 choose 3. A team-count larger than the arrange-count fails the opposite test.

Comparing 12 outfits to 5! is mute.

Figure. The opposite test: choosing 2 from 5 to keep is the same count as choosing 3 to leave behind. 5C2 = 5C3 = 10, so the two bars match.

How it works

  1. State 60 or 10 or 12The claim.
  2. Rebuild the falling product or divide by r!The check.
  3. Mend if they disagreeHonest.
5 people, 3 chairs (order matters) is
  1. 60 — 5×4×3
  2. 10
  3. 15

P(5,3); 10 would be the team.

Notes

  • The official chapter title is “Permutations and Combinations”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • P(n,r)=n!/(n−r)!
  • C(n,r)=n!/(r!(n−r)!)
  • sequence of jobs: multiply

Recap

Hold these pegs from the official chapter “Permutations and Combinations”. The wording is ExamMaster’s teaching, not a textbook recap.

Fundamental Principle of Counting
The fundamental counting principle: if one job has m ways and a next independent job has n ways, together there are m × n ways.
Permutations
A permutation is an arrangement — order matters.
Combinations
A combination is a team — order does not matter.
A definition is a test you can run
Permute / combine is a test: order-matters or not, plus the factorial write.
Name the given before the unknown
The given is n, r, and “arrange or team”. The unknown is the count. Copy “order matters” before you pick C.
One worked case is enough at this class
One count is enough. A page of the same 12-outfit clone does not add a new C(5,3).

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