Greatest Binomial Coefficient
On this page
Direct answer
The coefficients of (1 + x)^n climb to a crest and fall back symmetrically: since C(n, r+1)/C(n, r) = (n − r)/(r + 1), they rise while r < (n − 1)/2 and decline after. So the greatest binomial coefficient is C(n, n/2) alone when n is even, and the equal pair C(n, (n−1)/2) = C(n, (n+1)/2) when n is odd — a property of Pascal's row only, with no dependence on x. The numerically greatest term of an expansion, by contrast, depends on |y/x| and is located by a different ratio test; conflating the two is the chapter's most rewarded confusion.
What you must remember
- Monotonicity: C(n, r+1)/C(n, r) = (n − r)/(r + 1); coefficients increase while r < (n − 1)/2 and decrease afterwards — no full row needs computing.
- Greatest coefficient: n even → C(n, n/2), e.g. C(10, 5) = 252 for (1 + x)^10; n odd → both C(n, (n−1)/2) and C(n, (n+1)/2), e.g. C(15, 7) = C(15, 8) = 6435.
- Symmetry: C(n, r) = C(n, n − r); the row reads identically forwards and backwards, which is exactly why odd n produces a tie at the crest.
- Greatest term (the contrast): in (1 + x)^n with x > 0, T_(r+1) ≥ T_r ⟺ [(n − r + 1)/r]·x ≥ 1; the answer moves with x and can sit far from the middle.
- Row totals: Σ C(n, r) = 2^n and Σ (−1)^r C(n, r) = 0; the greatest coefficient is the largest summand feeding into 2^n.
- Values worth carrying: C(10, 5) = 252, C(12, 6) = 924, C(14, 7) = 3432, C(15, 7) = 6435 — these reappear as options constantly.
- Link to middle term: the middle term of (x + y)^n carries the greatest binomial coefficient whatever x and y are; only its numerical size as a term depends on the ratio.
Greatest coefficient versus greatest term
Find the greatest coefficient in (1 + x)^15, then the numerically greatest term in (1 + 2x)^15. First part: n odd, so the maxima are C(15, 7) and C(15, 8), each 6435 — positions 8 and 9 in the expansion. Second part: T(r+1) = C(15, r)·2^r, and the ratio test gives T(r+1)/T_r = [(16 − r)/r]·2 ≥ 1 ⟺ 32 − 2r ≥ r ⟺ r ≤ 32/3. So terms grow up to r = 10 and shrink after: the maximum sits at T₁₁ = C(15, 10)·2^10 = 3003 × 1024 = 3075072, the eleventh term — nowhere near the middle. The coefficient question read only Pascal's row; the term question multiplied the row by 2^r, and that exponential weighting dragged the peak to the right.
The lesson generalises: for (1 + kx)^n the greatest term drifts towards the end as k grows, while the greatest coefficient never moves. Whenever an option set offers "the middle term is the greatest", read carefully whether coefficient or term is meant — both statements can be true or false depending on that word.
Three questions that look identical
Greatest coefficient (parity of n alone), greatest term (depends on |y/x| through the ratio test), and middle term (position by parity) form Advanced's favourite triple — all three can appear in one multiple-correct item. Main dresses them up with substitutions: asking about (2 + x)^15 quietly re-indexes to (1 + x/2)^15 for coefficient questions, where the answer is still parity arithmetic. Traps: reporting one maximum where two exist (odd n always gives a pair); applying the coefficient rule to (2 + 3x)^n as if the row of Pascal still decided it — with constants attached, the "coefficients" include powers of 2 and 3 and the greatest-term logic returns; and ratio slips, using (n − r)/(r + 1) with an x multiplied in when the question was purely about coefficients.
Frequently asked questions
What is the greatest coefficient in (1 + x)^15?
Both C(15, 7) and C(15, 8), each equal to 6435 — odd n always yields two equal maxima at the middle.
Does the greatest coefficient depend on x?
No — it is a property of Pascal's row for n alone; the numerically greatest term is the quantity that depends on x.
How do I locate the numerically greatest term in (1 + x)^n?
Solve [(n − r + 1)/r]·|x| ≥ 1 for the largest admissible r; the maximum term is then T_(r+1).
Why are the two middle coefficients equal for odd n?
Symmetry C(n, r) = C(n, n − r) maps (n − 1)/2 onto (n + 1)/2, so the twin crests match exactly.
Is the middle term of (x + y)^n the one with the greatest coefficient?
Yes — the middle term always carries the greatest binomial coefficient, though not necessarily the numerically greatest term when |y/x| ≠ 1.