{
  "index": "1992-B-2",
  "type": "COMB",
  "tag": [
    "COMB",
    "ALG"
  ],
  "difficulty": "",
  "question": "the coefficient of $x^k$ in the expansion of $(1 + x + x^2 + x^3)^n$.\nProve that\n\\[\nQ(n, k) = \\sum_{j=0}^k \\binom{n}{j} \\binom{n}{k-2j},\n\\]\nwhere $\\binom{a}{b}$ is the standard binomial coefficient. (Reminder: For\nintegers $a$ and $b$ with $a \\geq 0$, $\\binom{a}{b} =\n\\frac{a!}{b!(a-b)!}$ for $0 \\leq b \\leq a$, with $\\binom{a}{b} = 0$ otherwise.)",
  "solution": "Solution. Write \\( \\left(1+x+x^{2}+x^{3}\\right)^{n} \\) as \\( \\left(1+x^{2}\\right)^{n}(1+x)^{n} \\). The coefficient of \\( x^{k} \\) gets contributions from the \\( x^{2 j} \\) term in the first factor (with coefficient \\( \\binom{n}{j} \\) ) times the \\( x^{k-2 j} \\) term in the second factor (with coefficient \\( \\binom{n}{k-2 j} \\) ).",
  "vars": [
    "x",
    "j"
  ],
  "params": [
    "k",
    "n",
    "a",
    "b"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "indetermin",
        "j": "indexer",
        "k": "targetexp",
        "n": "powercount",
        "a": "paramalpha",
        "b": "parambeta"
      },
      "question": "the coefficient of $indetermin^{targetexp}$ in the expansion of $(1 + indetermin + indetermin^{2} + indetermin^{3})^{powercount}$. Prove that\n\\[\nQ(powercount, targetexp) = \\sum_{indexer=0}^{targetexp} \\binom{powercount}{indexer} \\binom{powercount}{targetexp-2 indexer},\n\\]\nwhere $\\binom{paramalpha}{parambeta}$ is the standard binomial coefficient. (Reminder: For\nintegers $paramalpha$ and $parambeta$ with $paramalpha \\geq 0$, $\\binom{paramalpha}{parambeta} =\n\\frac{paramalpha!}{parambeta!(paramalpha-parambeta)!}$ for $0 \\leq parambeta \\leq paramalpha$, with $\\binom{paramalpha}{parambeta} = 0$ otherwise.)",
      "solution": "Solution. Write \\( \\left(1+indetermin+indetermin^{2}+indetermin^{3}\\right)^{powercount} \\) as \\( \\left(1+indetermin^{2}\\right)^{powercount}(1+indetermin)^{powercount} \\). The coefficient of \\( indetermin^{targetexp} \\) gets contributions from the \\( indetermin^{2 indexer} \\) term in the first factor (with coefficient \\( \\binom{powercount}{indexer} \\) ) times the \\( indetermin^{targetexp-2 indexer} \\) term in the second factor (with coefficient \\( \\binom{powercount}{targetexp-2 indexer} \\) )."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "meadowgrass",
        "j": "sandcastle",
        "k": "hummingbird",
        "n": "thunderbolt",
        "a": "parchment",
        "b": "lighthouse"
      },
      "question": "the coefficient of $meadowgrass^{hummingbird}$ in the expansion of $(1 + meadowgrass + meadowgrass^2 + meadowgrass^3)^{thunderbolt}$. Prove that\n\\[\nQ(thunderbolt, hummingbird) = \\sum_{sandcastle=0}^{hummingbird} \\binom{thunderbolt}{sandcastle} \\binom{thunderbolt}{hummingbird-2 sandcastle},\n\\]\nwhere $\\binom{parchment}{lighthouse}$ is the standard binomial coefficient. (Reminder: For\nintegers $parchment$ and $lighthouse$ with $parchment \\geq 0$, $\\binom{parchment}{lighthouse} =\n\\frac{parchment!}{lighthouse!(parchment-lighthouse)!}$ for $0 \\leq lighthouse \\leq parchment$, with $\\binom{parchment}{lighthouse} = 0$ otherwise.)",
      "solution": "Solution. Write \\( \\left(1+meadowgrass+meadowgrass^{2}+meadowgrass^{3}\\right)^{thunderbolt} \\) as \\( \\left(1+meadowgrass^{2}\\right)^{thunderbolt}(1+meadowgrass)^{thunderbolt} \\). The coefficient of \\( meadowgrass^{hummingbird} \\) gets contributions from the \\( meadowgrass^{2 sandcastle} \\) term in the first factor (with coefficient \\( \\binom{thunderbolt}{sandcastle} \\) ) times the \\( meadowgrass^{hummingbird-2 sandcastle} \\) term in the second factor (with coefficient \\( \\binom{thunderbolt}{hummingbird-2 sandcastle} \\) )."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantvalue",
        "j": "staticindex",
        "k": "basenumber",
        "n": "fractionpart",
        "a": "irrational",
        "b": "continuous"
      },
      "question": "the coefficient of $constantvalue^{basenumber}$ in the expansion of $(1 + constantvalue + constantvalue^2 + constantvalue^3)^{fractionpart}$. Prove that\n\\[\nQ(fractionpart, basenumber) = \\sum_{staticindex=0}^{basenumber} \\binom{fractionpart}{staticindex} \\binom{fractionpart}{basenumber-2staticindex},\n\\]\nwhere $\\binom{irrational}{continuous}$ is the standard binomial coefficient. (Reminder: For\nintegers $irrational$ and $continuous$ with $irrational \\geq 0$, $\\binom{irrational}{continuous} =\n\\frac{irrational!}{continuous!(irrational-continuous)!}$ for $0 \\leq continuous \\leq irrational$, with $\\binom{irrational}{continuous} = 0$ otherwise.)",
      "solution": "Solution. Write \\( \\left(1+constantvalue+constantvalue^{2}+constantvalue^{3}\\right)^{fractionpart} \\) as \\( \\left(1+constantvalue^{2}\\right)^{fractionpart}(1+constantvalue)^{fractionpart} \\). The coefficient of \\( constantvalue^{basenumber} \\) gets contributions from the \\( constantvalue^{2 staticindex} \\) term in the first factor (with coefficient \\( \\binom{fractionpart}{staticindex} \\) ) times the \\( constantvalue^{basenumber-2 staticindex} \\) term in the second factor (with coefficient \\( \\binom{fractionpart}{basenumber-2 staticindex} \\) )."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "j": "hjgrksla",
        "k": "pyfmrueo",
        "n": "zqdvsekl",
        "a": "mthgsxco",
        "b": "rcnlpkga"
      },
      "question": "the coefficient of $qzxwvtnp^{pyfmrueo}$ in the expansion of $(1 + qzxwvtnp + qzxwvtnp^2 + qzxwvtnp^3)^{zqdvsekl}$. Prove that\n\\[\nQ(zqdvsekl, pyfmrueo) = \\sum_{hjgrksla=0}^{pyfmrueo} \\binom{zqdvsekl}{hjgrksla} \\binom{zqdvsekl}{pyfmrueo-2hjgrksla},\n\\]\nwhere $\\binom{mthgsxco}{rcnlpkga}$ is the standard binomial coefficient. (Reminder: For\nintegers $mthgsxco$ and $rcnlpkga$ with $mthgsxco \\geq 0$, $\\binom{mthgsxco}{rcnlpkga} =\n\\frac{mthgsxco!}{rcnlpkga!(mthgsxco-rcnlpkga)!}$ for $0 \\leq rcnlpkga \\leq mthgsxco$, with $\\binom{mthgsxco}{rcnlpkga} = 0$ otherwise.)",
      "solution": "Solution. Write \\( \\left(1+qzxwvtnp+qzxwvtnp^{2}+qzxwvtnp^{3}\\right)^{zqdvsekl} \\) as \\( \\left(1+qzxwvtnp^{2}\\right)^{zqdvsekl}(1+qzxwvtnp)^{zqdvsekl} \\). The coefficient of \\( qzxwvtnp^{pyfmrueo} \\) gets contributions from the \\( qzxwvtnp^{2 hjgrksla} \\) term in the first factor (with coefficient \\( \\binom{zqdvsekl}{hjgrksla} \\) ) times the \\( qzxwvtnp^{pyfmrueo-2 hjgrksla} \\) term in the second factor (with coefficient \\( \\binom{zqdvsekl}{pyfmrueo-2 hjgrksla} \\) )."
    },
    "kernel_variant": {
      "question": "Let n,k ∈ ℕ.  Determine the coefficient of xᵏ in  \n\n P(x) = (1 + x³ + x⁷ + x¹⁰ + x¹⁵ + x¹⁸ + x²² + x²⁵ + x³¹ +\n            x³⁴ + x³⁸ + x⁴¹ + x⁴⁶ + x⁴⁹ + x⁵³ + x⁵⁶)ⁿ.  \n\nProve that this coefficient equals  \n\n T(n,k) =  Σ_{a=0}^{⌊k/31⌋}  Σ_{b=0}^{⌊(k−31a)/15⌋}  Σ_{c=0}^{⌊(k−31a−15b)/7⌋}\n                     C(n,a) C(n,b) C(n,c) C(n, (k−31a−15b−7c)/3 ),\n\nwith the usual convention C(n,m)=0 whenever m∉ℕ or m>n.",
      "solution": "Step 1.  Hidden factorisation of the base polynomial.  \nWrite  \n\n 1 + x³ + x⁷ + x¹⁰ = (1 + x³)(1 + x⁷),  \n 1 + x¹⁵ + x¹⁸ + x²² + x²⁵ = (1 + x¹⁵)(1 + x³ + x⁷ + x¹⁰)     \n                 = (1 + x³)(1 + x⁷)(1 + x¹⁵),  \n\nso  \n\n 1 + x³ + x⁷ + x¹⁰ + x¹⁵ + x¹⁸ + x²² + x²⁵ = (1 + x³)(1 + x⁷)(1 + x¹⁵).\n\nMultiplying this by (1 + x³¹) doubles the number of terms, giving exactly the\n16 monomials displayed in P(x):\n\n (1 + x³)(1 + x⁷)(1 + x¹⁵)(1 + x³¹)\n  = 1 + x³ + x⁷ + x¹⁰ + x¹⁵ + x¹⁸ + x²² + x²⁵\n    + x³¹ + x³⁴ + x³⁸ + x⁴¹ + x⁴⁶ + x⁴⁹ + x⁵³ + x⁵⁶ .\n\nHence  \n\n P(x) = [(1 + x³)(1 + x⁷)(1 + x¹⁵)(1 + x³¹)]ⁿ .     (1)\n\nStep 2.  Pulling the power inside.  \nBecause the ring ℤ[x] is commutative, from (1)\n\n P(x) = (1 + x³)ⁿ (1 + x⁷)ⁿ (1 + x¹⁵)ⁿ (1 + x³¹)ⁿ.     (2)\n\nStep 3.  Expanding each factor.  \nFor j = 0,…,n we have\n\n (1 + x^d)ⁿ = Σ_{j=0}^{n} C(n,j) x^{dj} , d ∈ {3,7,15,31}.    (3)\n\nMultiplying the four expansions in (2) and collecting x–powers, the exponent k\narises exactly from quadruples (j₁,j₂,j₃,j₄) with\n\n 3j₁ + 7j₂ + 15j₃ + 31j₄ = k, 0 ≤ j₁,j₂,j₃,j₄ ≤ n.     (4)\n\nFor such a quadruple the product of the binomial coefficients\n\n C(n,j₁) C(n,j₂) C(n,j₃) C(n,j₄)\n\nis the contribution to the coefficient of xᵏ.  Summing over all admissible\nquadruples yields the desired coefficient.\n\nStep 4.  Reducing the diophantine system.  \nFix j₄=a, j₃=b, j₂=c successively.\n\n•  Choose a ≥0 with 31a ≤ k.  \n•  With a fixed, choose b ≥0 with 15b ≤ k–31a.  \n•  With a,b fixed, choose c ≥0 with 7c ≤ k–31a–15b.  \n•  Finally j₁ is forced to be  \n\n j₁ = (k – 31a – 15b – 7c)/3.     (5)\n\nEquation (4) guarantees that (5) is an integer; if it is not, or if j₁>n,\nthe corresponding term is interpreted as 0 via the binomial-coefficient\nconvention.  Therefore the coefficient of xᵏ equals\n\n T(n,k) = Σ_{a=0}^{⌊k/31⌋} Σ_{b=0}^{⌊(k−31a)/15⌋} Σ_{c=0}^{⌊(k−31a−15b)/7⌋}\n        C(n,a) C(n,b) C(n,c) C(n, (k−31a−15b−7c)/3) ,\n\nestablishing the claimed formula. ∎",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.725286",
        "was_fixed": false,
        "difficulty_analysis": "•  Extra structural layer: the original problem required splitting one factor\n   into two parts; the current kernel variant required three.  The enhanced\n   problem needs a four-factor decomposition, leading to a\n   four-parameter Diophantine constraint.  \n\n•  Higher combinatorial dimension: the answer involves a triple\n   summation (over a,b,c) instead of the single or double sums in the previous\n   versions.  Handling the fourth implicit parameter demands careful control\n   of integrality and bounds.  \n\n•  Larger search space: the base polynomial has 16 distinct monomials (versus\n   8 in the kernel variant and 4 in the original), so naïve\n   coefficient-extraction by enumeration becomes dramatically more difficult.\n   The factorisation trick is indispensable but much less obvious.  \n\n•  Deeper insight: one must recognise a concealed binary-expansion structure\n   {3,7,15,31} permitting the fourfold factorisation and\n   systematically translate it into a linear Diophantine equation in four\n   variables, then manage the resulting nested summations.  \n\nHence the enhanced variant introduces significantly more technical\ncomplexity, requires a higher-dimensional combinatorial argument, and cannot\nbe solved by straightforward pattern matching."
      }
    },
    "original_kernel_variant": {
      "question": "Let n,k ∈ ℕ.  Determine the coefficient of xᵏ in  \n\n P(x) = (1 + x³ + x⁷ + x¹⁰ + x¹⁵ + x¹⁸ + x²² + x²⁵ + x³¹ +\n            x³⁴ + x³⁸ + x⁴¹ + x⁴⁶ + x⁴⁹ + x⁵³ + x⁵⁶)ⁿ.  \n\nProve that this coefficient equals  \n\n T(n,k) =  Σ_{a=0}^{⌊k/31⌋}  Σ_{b=0}^{⌊(k−31a)/15⌋}  Σ_{c=0}^{⌊(k−31a−15b)/7⌋}\n                     C(n,a) C(n,b) C(n,c) C(n, (k−31a−15b−7c)/3 ),\n\nwith the usual convention C(n,m)=0 whenever m∉ℕ or m>n.",
      "solution": "Step 1.  Hidden factorisation of the base polynomial.  \nWrite  \n\n 1 + x³ + x⁷ + x¹⁰ = (1 + x³)(1 + x⁷),  \n 1 + x¹⁵ + x¹⁸ + x²² + x²⁵ = (1 + x¹⁵)(1 + x³ + x⁷ + x¹⁰)     \n                 = (1 + x³)(1 + x⁷)(1 + x¹⁵),  \n\nso  \n\n 1 + x³ + x⁷ + x¹⁰ + x¹⁵ + x¹⁸ + x²² + x²⁵ = (1 + x³)(1 + x⁷)(1 + x¹⁵).\n\nMultiplying this by (1 + x³¹) doubles the number of terms, giving exactly the\n16 monomials displayed in P(x):\n\n (1 + x³)(1 + x⁷)(1 + x¹⁵)(1 + x³¹)\n  = 1 + x³ + x⁷ + x¹⁰ + x¹⁵ + x¹⁸ + x²² + x²⁵\n    + x³¹ + x³⁴ + x³⁸ + x⁴¹ + x⁴⁶ + x⁴⁹ + x⁵³ + x⁵⁶ .\n\nHence  \n\n P(x) = [(1 + x³)(1 + x⁷)(1 + x¹⁵)(1 + x³¹)]ⁿ .     (1)\n\nStep 2.  Pulling the power inside.  \nBecause the ring ℤ[x] is commutative, from (1)\n\n P(x) = (1 + x³)ⁿ (1 + x⁷)ⁿ (1 + x¹⁵)ⁿ (1 + x³¹)ⁿ.     (2)\n\nStep 3.  Expanding each factor.  \nFor j = 0,…,n we have\n\n (1 + x^d)ⁿ = Σ_{j=0}^{n} C(n,j) x^{dj} , d ∈ {3,7,15,31}.    (3)\n\nMultiplying the four expansions in (2) and collecting x–powers, the exponent k\narises exactly from quadruples (j₁,j₂,j₃,j₄) with\n\n 3j₁ + 7j₂ + 15j₃ + 31j₄ = k, 0 ≤ j₁,j₂,j₃,j₄ ≤ n.     (4)\n\nFor such a quadruple the product of the binomial coefficients\n\n C(n,j₁) C(n,j₂) C(n,j₃) C(n,j₄)\n\nis the contribution to the coefficient of xᵏ.  Summing over all admissible\nquadruples yields the desired coefficient.\n\nStep 4.  Reducing the diophantine system.  \nFix j₄=a, j₃=b, j₂=c successively.\n\n•  Choose a ≥0 with 31a ≤ k.  \n•  With a fixed, choose b ≥0 with 15b ≤ k–31a.  \n•  With a,b fixed, choose c ≥0 with 7c ≤ k–31a–15b.  \n•  Finally j₁ is forced to be  \n\n j₁ = (k – 31a – 15b – 7c)/3.     (5)\n\nEquation (4) guarantees that (5) is an integer; if it is not, or if j₁>n,\nthe corresponding term is interpreted as 0 via the binomial-coefficient\nconvention.  Therefore the coefficient of xᵏ equals\n\n T(n,k) = Σ_{a=0}^{⌊k/31⌋} Σ_{b=0}^{⌊(k−31a)/15⌋} Σ_{c=0}^{⌊(k−31a−15b)/7⌋}\n        C(n,a) C(n,b) C(n,c) C(n, (k−31a−15b−7c)/3) ,\n\nestablishing the claimed formula. ∎",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.563769",
        "was_fixed": false,
        "difficulty_analysis": "•  Extra structural layer: the original problem required splitting one factor\n   into two parts; the current kernel variant required three.  The enhanced\n   problem needs a four-factor decomposition, leading to a\n   four-parameter Diophantine constraint.  \n\n•  Higher combinatorial dimension: the answer involves a triple\n   summation (over a,b,c) instead of the single or double sums in the previous\n   versions.  Handling the fourth implicit parameter demands careful control\n   of integrality and bounds.  \n\n•  Larger search space: the base polynomial has 16 distinct monomials (versus\n   8 in the kernel variant and 4 in the original), so naïve\n   coefficient-extraction by enumeration becomes dramatically more difficult.\n   The factorisation trick is indispensable but much less obvious.  \n\n•  Deeper insight: one must recognise a concealed binary-expansion structure\n   {3,7,15,31} permitting the fourfold factorisation and\n   systematically translate it into a linear Diophantine equation in four\n   variables, then manage the resulting nested summations.  \n\nHence the enhanced variant introduces significantly more technical\ncomplexity, requires a higher-dimensional combinatorial argument, and cannot\nbe solved by straightforward pattern matching."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}