When order changes the outcome

Ask whether swapping two chosen people creates a different result. A captain and a deputy have different roles; a two-person team has no order. Three people A,B,C give six ordered pairs: AB,AC,BA,BC,CA,CB. They give three teams: A,B,A,C,B,C. Each team appeared 2! times in the ordered list. For successive choices multiply the numbers of available choices. For alternatives that cannot happen together, add their counts. State what each factor or case counts. n!=n(n-1). s2. 1. There is one way to arrange no objects, so 0!=1; this also preserves 1!=1. 0!. For r ordered places filled from n distinct objects without replacement, nP_r=n(n-1). s(n-r+1)= n!(n-r)!. For an unordered selection divide by the r! orders of each selection: nC_r= n!r!(n-r)!. The formula list supplies the combination formula as a binomial coefficient. Know how to form the permutation product for ordered choices. Identify the outcomes first; a formula cannot decide whether order matters. Cancel factorials by hand for Paper 1.