MathematicsChapter 6: Permutation and Combination

MIXED COUNTING AND nCr IDENTITIES

A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys.

Learning objectives

State the main ideas and notation used in mixed counting and ncr identities.
Apply the rules of mixed counting and ncr identities in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out'). Meaning: split on a condition, add the disjoint piles. Do not multiply piles that are alternatives.

Where / symbols

MIXED COUNTING AND nCr IDENTITIES: A team of 4 from 6 boys and 5 girls with at least one girl: split cases 1,2,3,4 girls, or complement: all teams minus 4 boys. Pascal: nCr + nC(r-1) = (n+1)Cr (school identity, 'a particular person in or out'). Meaning: split on a condition, add the disjoint piles. Do not multiply piles that are alternatives.
1. AT LEAST / AT MOST: At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.
2. PASCAL AND PATHS: A person X is either in the committee (then choose r-1 from n-1) or out (choose r from n-1). That proves the identity. Useful to compute a row of Pascal's triangle.

Relationship

Pascal: nCr + nC(r-1) = (n+1)Cr

Overview

Cases, then Pascal:

At Least / At Most

At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.

Examples All - none · Add k=0 to k=2 · Product inside a case if two types

Pascal and Paths

A person X is either in the committee (then choose r-1 from n-1) or out (choose r from n-1). That proves the identity. Useful to compute a row of Pascal's triangle.

Examples (n-1)C(r-1) + (n-1)Cr = nCr · Triangle of nCr · Check small numbers

WHAT — MIXED COUNTING AND nCr IDENTITIES

RULE — 1. AT LEAST / AT MOST

At least one = total - none. At most two = 0+1+2. Write each nCr. Keep boys and girls as two types.

METHOD — 2. PASCAL AND PATHS

A person X is either in the committee (then choose r-1 from n-1) or out (choose r from n-1). That proves the identity. Useful to compute a row of Pascal's triangle.

WORKED EXAMPLE

Worked: committees of 3 from 5 people, at least one of A or B (idea: complement). total 5C3 = 10 with neither A nor B: 3C3 = 1 so 10-1 = 9 have A or B or both

How to answer a mixed counting and ncr identities question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

committees of 3 from 5 people, at least one of A or B (idea: complement).
total 5C3 = 10
with neither A nor B: 3C3 = 1
so 10-1 = 9 have A or B or both

In simple words

At least one is not nC1. It is a sum of cases or a complement.

Key points

at least: cases or complement
Pascal: nCr + nC(r-1) = (n+1)Cr
complement is faster often
in or out

Exam preparation

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