Counting and the Binomial Theorem
Permutations, Combinations, and Binomial Expansion
Begin every counting problem by deciding whether order matters. If it does, use permutations: nPr = n!/(n − r)!. If it does not, use combinations: nCr = n!/(r!(n − r)!). The fundamental counting principle multiplies independent choices, and factorial counts the arrangements of n distinct objects.
Special cases matter. Arrangements of n objects with repeats a, b, … identical is n!/(a!b!…). Circular arrangements of n distinct objects number (n − 1)!. When a group must stay together, treat it as one block, then multiply by the internal arrangements of the block. For "at least one" conditions, count the complement and subtract from the total - usually far faster than casework.
The binomial theorem expands (a + b)ⁿ as a sum of terms; the general term is t_{k+1} = nCk · a^{n−k} · b^k. Note the index shift: the term containing b^k is the (k + 1)th term. Track signs carefully when b is negative, since (−b)^k alternates.
Traps: using permutations where order is irrelevant; forgetting to divide by factorials of identical items; ignoring that a block also has internal order; off-by-one in the general term; dropping a needed case in "at least" problems.
Worked example: A committee of 4 is chosen from 5 Grade 12 and 7 Grade 11 students and must include at least one Grade 12 student. Total committees: 12C4 = 495. Committees with no Grade 12 student: 7C4 = 35. Answer: 495 − 35 = 460.
Rule of thumb: order matters → permutation; order does not → combination; and "at least one" usually means total minus none.