Counting and probability

A combination is a selection from distinct objects in which order has no significance. A team containing Alex, Bo and Chen is the same team as Chen, Alex and Bo. Before calculating, identify the objects, the required number and whether roles distinguish the selected people. Choosing three team members uses combinations; choosing a captain, secretary and treasurer does not, because changing roles changes the outcome.

State this modelling decision explicitly rather than choosing a formula simply because the question contains the word choose.

Factorial notation records descending products: n! = n(n − 1)…1, with 0! = 1. The number of unordered selections of r objects from n distinct objects is written nCr or C(n,r), and equals n!/[r!(n − r)!]. The numerator counts arrangements, while division removes arrangements of the selected and unselected groups that do not change the selection. For small numbers, cancelling common factors before multiplying avoids unnecessary arithmetic.

A calculator combination command is useful for large inputs, but does not decide whether order matters.

Pascal’s triangle begins with the row 1, numbered zero. Each new row starts and ends with 1; each internal entry is the sum of the two entries immediately above it. Rows one through four are 1,1; 1,2,1; 1,3,3,1; and 1,4,6,4,1. Row n contains C(n,r) as r runs from zero to n. To expand (x + y)ⁿ, choose y from r of the n factors and x from all remaining factors. Thus the term xⁿ⁻ʳyʳ has coefficient C(n,r).