Coefficients
Two coefficient ideas dominate CAT-style binomial questions. First, the SUM of all coefficients in any polynomial expansion is found by substituting every variable = 1. So the sum of coefficients in (2x + 3y)ⁿ is just (2+3)ⁿ = 5ⁿ — no expansion needed. The pure binomial coefficients (the nCr values alone) sum to 2ⁿ, because that is (1+1)ⁿ; with alternating signs, (1−1)ⁿ = 0, so the alternate-sign sum is 0 for n ≥ 1. Second, the GREATEST coefficient: among nC0…nCn the values rise to the middle and fall back (they are symmetric), so the largest is the central one — nC(n/2) when n is even, and nC((n−1)/2) = nC((n+1)/2) (two equal greatest values) when n is odd. Careful: the "greatest coefficient" can mean the greatest numerical coefficient including the a and b factors, which may not sit at the centre — read the question. Substituting clever values (1, −1, 0) into the expansion is the master trick of this topic.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Expansion & general term
| Binomial expansion | (a+b)ⁿ = Σ nCr · a^(n−r) · b^r, r = 0…n |
|---|---|
| General term | T(r+1) = nCr · a^(n−r) · b^r |
| Number of terms | (a+b)ⁿ has (n+1) terms |
| Binomial coefficient | nCr = n! / [r!(n−r)!] |
| Symmetry of coefficients | nCr = nC(n−r) |
Middle term, sums & special cases
| Middle term (n even) | single term T(n/2 + 1) |
|---|---|
| Middle terms (n odd) | two terms T((n+1)/2) and T((n+3)/2) |
| Sum of all coefficients | put each variable = 1 ⇒ (sum of bases)ⁿ |
| Sum of binomial coefficients | nC0 + nC1 + … + nCn = 2ⁿ |
| Greatest coefficient | the middle coefficient: nC(n/2) (n even) |