Combinations
Combinations tests selection without regard to order, properties of nCr, committee problems, and counting geometric figures. The exam often contrasts 'select' with 'arrange'.
Core rules
nCr=n!/(r!(n−r)!) counts r-object subsets from n distinct objects. Key properties: nC0=nCn=1, nCr=nC(n−r), and nCr+nC(r−1)=n+1Cr. The number of subsets of an n-set is 2ⁿ; non-empty subsets are 2ⁿ−1. For committees, multiply choices from separate groups and add cases that satisfy 'at least' or 'at most'. For geometry, choose points and subtract collinear or degenerate selections.
Worked example
From 6 men and 5 women, form a committee of 4 with at least 2 women. Count cases: exactly 2 women: 5C2×6C2=10×15=150; exactly 3 women: 5C3×6C1=10×6=60; exactly 4 women: 5C4×6C0=5×1=5. Total=150+60+5=215.
Traps
Do not use nPr when the order in the committee does not matter. 'At least 2 women' includes 2, 3 and 4 women; do not stop at 2. In 'at most' problems, include zero if allowed. When choosing points for triangles, subtract sets of three collinear points. Avoid double counting cases split by category.