Permutations and Combinations

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Fundamental Principle of Counting

Part of: Counting and Permutations

Independent stages multiply: m ways then n ways give m-times-n; addition covers exclusive alternatives; digit, flag and word counts chain choices.

Why it matters. Multiplication rules every arrangement. Stages stack into formulas.

Formulas

m ways then n ways = m x n combined ways
Exclusive alternatives: m ways or n ways = m + n ways

Key words

Multiplication principle: : The rule that m ways then n ways give m x n combined ways.
Addition principle: : The rule that exclusive alternatives add: m ways or n ways give m + n ways.
Independent stages: : Steps whose choices do not affect each other, so their counts multiply.
Sample count: : The total from chaining choices across digits, flags or words.

Try it in two minutes

Count each stage.
Multiply the stages.
Add exclusive cases.

Try at home. Count outfits from shirts and pants.

Factorials and Permutations

Part of: Counting and Permutations

n-factorial multiplies one to n with zero-factorial one; nPr equals n-factorial over n-minus-r-factorial; full arrangements give n-factorial.

Why it matters. Factorials compact counting. nPr arranges selections.

Formulas

n! = 1 x 2 x ... x n
0! = 1
nPr = n! / (n - r)!
Full arrangement of n objects = n!

Key words

Factorial: : The product n! = 1 x 2 x ... x n, with 0! = 1.
Permutation: : An ordered arrangement of r distinct objects chosen from n.
nPr: : The count of ordered arrangements of r objects from n distinct objects.
Full arrangement: : The ordering of all n objects, counted by n!.

Try it in two minutes

Expand the factorial.
Divide by leftover factorial.
Fill slots left to right.

Try at home. Arrange 4 books on a shelf.

Permutations with Repetition and Restrictions

Part of: Restricted Permutations and Combinations

Repeated items divide by repeat-factorials; tied objects glue into one bundle; fixed ends shrink slots; gaps place separators apart.

Why it matters. Restrictions reshape counts. Bundling tames togetherness.

Formulas

Repeats: n! / (p1! x p2! x ... )
Bundle of k tied objects: arrange as (n - k + 1)! x k!

Key words

Repeated permutation: : An arrangement where identical items divide out repeat orderings.
Bundle method: : Gluing tied objects into one unit, arranging, then arranging inside.
Gap method: : Placing separators into gaps so kept-apart items never sit together.
Fixed position: : A slot already decided, which shrinks the remaining choices.

Try it in two minutes

Divide repeats out.
Bundle tied objects.
Fix special slots first.

Try at home. Arrange letters of LEVEL.

Combinations and Selection

Part of: Restricted Permutations and Combinations

nCr equals n-factorial over r-factorial n-minus-r-factorial; order-blindness divides nPr by r-factorial; symmetry pairs r with n-minus-r; committees split into cases.

Why it matters. Selections ignore order. Symmetry halves work.

Formulas

nCr = n! / (r! x (n - r)!)
nCr = nPr / r!
nCr = nC(n - r)

Key words

Combination: : A selection of r objects from n where order does not matter.
nCr: : The count of order-free groups of r objects chosen from n.
Symmetry of choice: : The pairing that choosing r to keep equals choosing n - r to leave out.
Committee cases: : Splitting a selection into exclusive cases and adding their counts.

Try it in two minutes

Divide arrangements by slot orders.
Pair r with n-minus-r.
Add committee cases.

Try at home. Choose 3 toppings from 8.

What your child will practise

Combinations and Selection: Desert Quick Fire (Quick Fire)
Combinations and Selection: Jungle Quest (Flip and Match)
Factorials and Permutations: Space Puzzle (Clue Puzzle)
Fundamental Principle of Counting: Ocean Quick Fire (Quick Fire)
Permutations with Repetition and Restrictions: Jungle Puzzle (Clue Puzzle)

Quiz: Combinations and Selection (8 questions)
Quiz: Factorials and Permutations (8 questions)
Quiz: Fundamental Principle of Counting (8 questions)
Quiz: Permutations with Repetition and Restrictions (8 questions)

Permutations and Combinations

Class 11 · Mathematics

4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.

Fundamental Principle of Counting

Part of: Counting and Permutations

Independent stages multiply: m ways then n ways give m-times-n; addition covers exclusive alternatives; digit, flag and word counts chain choices.

Why it matters. Multiplication rules every arrangement. Stages stack into formulas.

Formulas

m ways then n ways = m x n combined ways
Exclusive alternatives: m ways or n ways = m + n ways

Key words

Multiplication principle: : The rule that m ways then n ways give m x n combined ways.
Addition principle: : The rule that exclusive alternatives add: m ways or n ways give m + n ways.
Independent stages: : Steps whose choices do not affect each other, so their counts multiply.
Sample count: : The total from chaining choices across digits, flags or words.

Try it in two minutes

Count each stage.
Multiply the stages.
Add exclusive cases.

Try at home. Count outfits from shirts and pants.

Factorials and Permutations

Part of: Counting and Permutations

n-factorial multiplies one to n with zero-factorial one; nPr equals n-factorial over n-minus-r-factorial; full arrangements give n-factorial.

Why it matters. Factorials compact counting. nPr arranges selections.

Formulas

n! = 1 x 2 x ... x n
0! = 1
nPr = n! / (n - r)!
Full arrangement of n objects = n!

Key words

Factorial: : The product n! = 1 x 2 x ... x n, with 0! = 1.
Permutation: : An ordered arrangement of r distinct objects chosen from n.
nPr: : The count of ordered arrangements of r objects from n distinct objects.
Full arrangement: : The ordering of all n objects, counted by n!.

Try it in two minutes

Expand the factorial.
Divide by leftover factorial.
Fill slots left to right.

Try at home. Arrange 4 books on a shelf.

Permutations with Repetition and Restrictions

Part of: Restricted Permutations and Combinations

Repeated items divide by repeat-factorials; tied objects glue into one bundle; fixed ends shrink slots; gaps place separators apart.

Why it matters. Restrictions reshape counts. Bundling tames togetherness.

Formulas

Repeats: n! / (p1! x p2! x ... )
Bundle of k tied objects: arrange as (n - k + 1)! x k!

Key words

Repeated permutation: : An arrangement where identical items divide out repeat orderings.
Bundle method: : Gluing tied objects into one unit, arranging, then arranging inside.
Gap method: : Placing separators into gaps so kept-apart items never sit together.
Fixed position: : A slot already decided, which shrinks the remaining choices.

Try it in two minutes

Divide repeats out.
Bundle tied objects.
Fix special slots first.

Try at home. Arrange letters of LEVEL.

Combinations and Selection

Part of: Restricted Permutations and Combinations

nCr equals n-factorial over r-factorial n-minus-r-factorial; order-blindness divides nPr by r-factorial; symmetry pairs r with n-minus-r; committees split into cases.

Why it matters. Selections ignore order. Symmetry halves work.

Formulas

nCr = n! / (r! x (n - r)!)
nCr = nPr / r!
nCr = nC(n - r)

Key words

Combination: : A selection of r objects from n where order does not matter.
nCr: : The count of order-free groups of r objects chosen from n.
Symmetry of choice: : The pairing that choosing r to keep equals choosing n - r to leave out.
Committee cases: : Splitting a selection into exclusive cases and adding their counts.

Try it in two minutes

Divide arrangements by slot orders.
Pair r with n-minus-r.
Add committee cases.

Try at home. Choose 3 toppings from 8.

What your child will practise

Combinations and Selection: Desert Quick Fire (Quick Fire)
Combinations and Selection: Jungle Quest (Flip and Match)
Factorials and Permutations: Space Puzzle (Clue Puzzle)
Fundamental Principle of Counting: Ocean Quick Fire (Quick Fire)
Permutations with Repetition and Restrictions: Jungle Puzzle (Clue Puzzle)

Quiz: Combinations and Selection (8 questions)
Quiz: Factorials and Permutations (8 questions)
Quiz: Fundamental Principle of Counting (8 questions)
Quiz: Permutations with Repetition and Restrictions (8 questions)