6. Permutations and Combinations

7 topics2 h

Chapter 6 of NCERT Mathematics for Class 11: Permutations and Combinations. Three sections — Counting without writing the list out, Arrangements, where order counts and Selections, where order does not. Seven videos, 2 h in all.

Worked answers to the 42 exercise questions in this chapter: Exercise 6.1 · Exercise 6.2 · Exercise 6.3 · Exercise 6.4 · Miscellaneous Exercise

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Part 1

Counting without writing the list out

12 minWhy choices made one after another multiply rather than addThree trousers and two shirts do not make five outfits. List every pairing and there are six — three equal groups of two, because equal groups are what multiplication counts.12 min · 12 parts

Part 2

Arrangements, where order counts

13 minFilling a row of places one at a time from a supply of distinct objectsFill three places from six letters, none repeated: six times five times four, a hundred and twenty ways. The last factor is n minus r plus one, not n minus r.13 min · 12 parts14 minA shorthand for descending products, and why the empty product is set to oneOne factorial equals one times zero factorial — that single line forces zero factorial to equal one, not zero, the way an empty sum is forced to equal nothing.14 min · 14 parts16 minThe closed formula, and how allowing repeats changes the count entirelySix times five times four, and six-factorial-over-three-factorial, are the same number reached two ways. The rewritten form changes nothing about the count, only what cancels.16 min · 15 parts14 minDividing out the swaps you cannot see when some objects are identicalLabel the two O's in ROOT and there are twenty-four arrangements of the four letters. Remove the labels and only twelve words remain — each spelled by exactly two.14 min · 14 parts

Part 3

Selections, where order does not

13 minEvery selection was counted once per arrangement of itselfThree players, two chosen for a team: counting with order gives six rows, but X-then-Y and Y-then-X are one team wearing two names. The rows fall into three equal pairs.13 min · 14 parts13 minChoosing what to leave out, and the rule that builds each count from two smaller onesCircling three objects out of five, in the same movement, names the two left behind. Choosing three and choosing two are not two counts but one, read from opposite ends.13 min · 13 parts

What you will be able to do

  • State the counting principle for two stages and for any finite number of stages
  • State what an arrangement in a fixed order is, and say why using only some of the objects still counts as one
  • Write out n! for small n and read the symbol aloud correctly
  • Reproduce the derivation that turns the descending product into a quotient of factorials
  • Explain why the plain factorial overcounts when two objects cannot be told apart
  • Recognise a question in which reordering the chosen objects does not produce a new answer
  • State the identity relating a selection count to the count of the complementary size, and justify it by describing a single act two ways

What this chapter assumes you already know

  • Multiplication as repeated addition of equal groups
  • Reading a branching diagram: a node, its branches, and the paths from root to tip
  • The idea that two descriptions can name the same object, so a list can double-count
  • Powers written as repeated multiplication, since the repeats-allowed count becomes one
  • That the count assigned to each stage must be a fixed number, independent of what earlier stages chose
  • Substituting particular whole numbers into an expression written with n and r
  • Reading a statement of the form "for all n and r satisfying a stated condition"
  • Cancelling a common factor from a numerator and a denominator

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Three pants and two shirts make five outfits."
  • "So counting problems are always multiplication."
  • "You can add the four signal counts because they are all signals."
  • "The factors just go down by one each time, always."
  • "Fill the places left to right."
  • "The tree proves it."
  • "m × n counts unordered pairings."
  • "nPr means n × r."