Counting Principles and Binomial Theorem
Counting principles and the binomial theorem
Use the addition principle for mutually exclusive cases and the multiplication principle for successive choices. Distinguish permutations, where order matters, from combinations, where it does not; the binomial coefficient C(n,k) counts k-element subsets. Handle restrictions by direct construction, complementary counting, or inclusion–exclusion. For the binomial expansion, the general term of (a+b)ⁿ is C(n,k)a^{n−k}b^k, and the sum of all binomial coefficients is 2ⁿ.
Traps: multiplying when cases should be added; forgetting to divide by repeated factorials for identical objects; confusing “at least” with “exactly”; and reporting a binomial coefficient when the question asks for the full coefficient, including powers of constants. For a constant term, set the exponent of the variable to zero and solve for k, then substitute.
Worked example: find the constant term in (x+2/x)⁶. The general term is C(6,k)x^{6−k}(2/x)^k=C(6,k)2^k x^{6−2k}. For a constant term, 6−2k=0, so k=3, and the term is C(6,3)2³=20·8=160. Rule of thumb: write the general term first; it turns coefficient, constant-term, and specific-power questions into one equation for k.