Properties of Binomial Coefficients: Identities Proved

Binomial coefficients are the numbers C(n, r) = n! / (r! (n − r)!) that appear in the expansion of (a + b)^n. They satisfy a small set of elegant identities — symmetry, sum rules, and Pascal's rule — that let you prove results and simplify expressions without computing factorials.

These identities matter because they turn messy coefficient sums into single powers of 2, connect neighbouring rows of Pascal's triangle, and appear as the counting argument behind countless combinatorics and probability proofs.

Key Concepts

• Binomial coefficient C(n, r): defined for integers 0 ≤ r ≤ n as n! / (r! (n − r)!).
• Symmetry: choosing r items is the same as leaving out n − r items.
• Row sums: adding all coefficients of one row of Pascal's triangle.
• Pascal's rule: each interior entry is the sum of the two entries above it.

Formulas

• Endpoints: C(n, 0) = C(n, n) = 1, since 0! = 1.
• Symmetry: C(n, r) = C(n, n − r), valid for 0 ≤ r ≤ n.
• Sum of all coefficients: C(n, 0) + C(n, 1) + … + C(n, n) = 2^n. Proof: put x = 1 in (1 + x)^n.
• Alternating sum: C(n, 0) − C(n, 1) + C(n, 2) − … + (−1)^nC(n, n) = 0, for n ≥ 1. Proof: put x = −1 in (1 + x)^n.
• Pascal's rule: C(n, r) + C(n, r − 1) = C(n + 1, r), valid for 1 ≤ r ≤ n.

Worked Example

Prove that the sum of the coefficients in even positions equals 2^n−1, i.e. C(n, 0) + C(n, 2) + C(n, 4) + … = 2^n−1. From the formulas above: (1 + 1)^n = 2^n gives the sum of all coefficients, and (1 − 1)^n = 0 gives the alternating sum. Adding these two equations cancels the odd-positioned terms and doubles the even-positioned ones: 2(C(n, 0) + C(n, 2) + …) = 2^n, so the even sum is 2^n−1. Check with n = 4: C(4, 0) + C(4, 2) + C(4, 4) = 1 + 6 + 1 = 8 = 2^3. Correct.

Common Misconceptions

• "The largest coefficient is always unique." Wrong — when n is odd there are two equal largest coefficients in the middle, e.g. C(5, 2) = C(5, 3) = 10.
• "The sum of the coefficients is 2^n − 1." Wrong — it is exactly 2^n; the −1 version appears only when you exclude one of the endpoint terms.
• "Symmetry means C(n, r) = C(r, n)." Wrong — symmetry swaps r with n − r, not n with r. C(n, r) = C(n, n − r) only.

Real-World Applications

• Combinatorics: counting arguments that sum selections over cases, as in "choose any number of items".
• Probability: total probability sums over all outcomes of binomial experiments.
• Algebra: simplifying long coefficient sums into closed forms.
• Computer science: analysing sums that arise in algorithm complexity.

The identities of binomial coefficients — symmetry, the 2^n row sum, the zero alternating sum, and Pascal's rule — compress long sums into short answers. Prove each once from (1 + x)^n, and you will recognise them instantly wherever coefficients appear.