{
  "index": "1985-A-1",
  "type": "COMB",
  "tag": [
    "COMB",
    "ALG"
  ],
  "difficulty": "",
  "question": "Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$\nof sets which have the property that\n\\begin{enumerate}\n\\item[(i)] $A_1 \\cup A_2 \\cup A_3 = \\{1,2,3,4,5,6,7,8,9,10\\}$, and\n\\item[(ii)] $A_1 \\cap A_2 \\cap A_3 = \\emptyset$.\n\\end{enumerate}\nExpress your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$\nare nonnegative integers.",
  "solution": "Solution. There is a bijection between triples of subsets of \\( \\{1, \\ldots, 10\\} \\) and \\( 10 \\times 3 \\) matrices with 0,1 entries, sending \\( \\left(A_{1}, A_{2}, A_{3}\\right) \\) to the matrix \\( B=\\left(b_{i j}\\right) \\) with \\( b_{i j}=1 \\) if \\( i \\in A_{j} \\) and \\( b_{i j}=0 \\) otherwise. Under this bijection the set \\( S \\) of triples satisfying\n\\[\nA_{1} \\cup A_{2} \\cup A_{3}=\\{1, \\ldots, 10\\} \\quad \\text { and } \\quad A_{1} \\cap A_{2} \\cap A_{3}=\\emptyset\n\\]\nmaps onto the set \\( T \\) of \\( 10 \\times 3 \\) matrices with 0,1 entries such that no row is (000) or (111). The number of possibilities for each row of such a matrix is \\( 2^{3}-2=6 \\), so \\( \\# T=6^{10} \\). Hence \\( \\# S=\\# T=2^{10} 3^{10} \\).\n\nReinterpretation. Equivalently, this problem asks for the number of ways of placing the numbers 1 through 10 in the Venn diagram of Figure 1, where no numbers are placed in the two regions marked with an \" \\( \\times \\) \".",
  "vars": [
    "A_1",
    "A_2",
    "A_3",
    "B",
    "b_ij",
    "i",
    "j",
    "S",
    "T"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "A_1": "setalpha",
        "A_2": "setbeta",
        "A_3": "setgamma",
        "B": "matrixb",
        "b_ij": "entrybij",
        "i": "rowidx",
        "j": "colidx",
        "S": "setspace",
        "T": "matspace"
      },
      "question": "Determine, with proof, the number of ordered triples $(setalpha, setbeta, setgamma)$ of sets which have the property that\n\\begin{enumerate}\n\\item[(i)] $setalpha \\cup setbeta \\cup setgamma = \\{1,2,3,4,5,6,7,8,9,10\\}$, and\n\\item[(ii)] $setalpha \\cap setbeta \\cap setgamma = \\emptyset$.\n\\end{enumerate}\nExpress your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.",
      "solution": "Solution. There is a bijection between triples of subsets of \\( \\{1, \\ldots, 10\\} \\) and \\( 10 \\times 3 \\) matrices with 0,1 entries, sending \\( \\left(setalpha, setbeta, setgamma\\right) \\) to the matrix \\( matrixb=\\left(entrybij\\right) \\) with \\( entrybij=1 \\) if \\( rowidx \\) is an element of the \\( colidx\\)-th set of the triple and \\( entrybij=0 \\) otherwise. Under this bijection the set \\( setspace \\) of triples satisfying\n\\[\nsetalpha \\cup setbeta \\cup setgamma=\\{1, \\ldots, 10\\} \\quad \\text { and } \\quad setalpha \\cap setbeta \\cap setgamma=\\emptyset\n\\]\nmaps onto the set \\( matspace \\) of \\( 10 \\times 3 \\) matrices with 0,1 entries such that no row is (000) or (111). The number of possibilities for each row of such a matrix is \\( 2^{3}-2=6 \\), so \\( \\# matspace=6^{10} \\). Hence \\( \\# setspace=\\# matspace=2^{10} 3^{10} \\).\n\nReinterpretation. Equivalently, this problem asks for the number of ways of placing the numbers 1 through 10 in the Venn diagram of Figure 1, where no numbers are placed in the two regions marked with an \" \\( \\times \\) \"."
    },
    "descriptive_long_confusing": {
      "map": {
        "A_1": "featherbed",
        "A_2": "lanternfish",
        "A_3": "blacksmith",
        "B": "raincloud",
        "b_ij": "dragonfly",
        "i": "paperclip",
        "j": "toadstool",
        "S": "gemstone",
        "T": "woodpecker"
      },
      "question": "Determine, with proof, the number of ordered triples $(featherbed, lanternfish, blacksmith)$\nof sets which have the property that\n\\begin{enumerate}\n\\item[(i)] $featherbed \\cup lanternfish \\cup blacksmith = \\{1,2,3,4,5,6,7,8,9,10\\}$, and\n\\item[(ii)] $featherbed \\cap lanternfish \\cap blacksmith = \\emptyset$.\n\\end{enumerate}\nExpress your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$\nare nonnegative integers.",
      "solution": "Solution. There is a bijection between triples of subsets of \\( \\{1, \\ldots, 10\\} \\) and \\( 10 \\times 3 \\) matrices with 0,1 entries, sending \\( \\left(featherbed, lanternfish, blacksmith\\right) \\) to the matrix \\( raincloud=\\left(dragonfly\\right) \\) with \\( dragonfly=1 \\) if \\( paperclip \\in A_{toadstool} \\) and \\( dragonfly=0 \\) otherwise. Under this bijection the set \\( gemstone \\) of triples satisfying\n\\[\nfeatherbed \\cup lanternfish \\cup blacksmith=\\{1, \\ldots, 10\\} \\quad \\text { and } \\quad featherbed \\cap lanternfish \\cap blacksmith=\\emptyset\n\\]\nmaps onto the set \\( woodpecker \\) of \\( 10 \\times 3 \\) matrices with 0,1 entries such that no row is (000) or (111). The number of possibilities for each row of such a matrix is \\( 2^{3}-2=6 \\), so \\( \\# woodpecker=6^{10} \\). Hence \\( \\# gemstone=\\# woodpecker=2^{10} 3^{10} \\).\n\nReinterpretation. Equivalently, this problem asks for the number of ways of placing the numbers 1 through 10 in the Venn diagram of Figure 1, where no numbers are placed in the two regions marked with an \" \\( \\times \\) \"."
    },
    "descriptive_long_misleading": {
      "map": {
        "A_1": "voidgroup",
        "A_2": "emptycluster",
        "A_3": "nullfamily",
        "B": "scalarbody",
        "b_ij": "bulkwhole",
        "i": "colmarker",
        "j": "rowmarker",
        "S": "singularone",
        "T": "monomatrix"
      },
      "question": "Determine, with proof, the number of ordered triples $(voidgroup, emptycluster, nullfamily)$\nof sets which have the property that\n\\begin{enumerate}\n\\item[(i)] $voidgroup \\cup emptycluster \\cup nullfamily = \\{1,2,3,4,5,6,7,8,9,10\\}$, and\n\\item[(ii)] $voidgroup \\cap emptycluster \\cap nullfamily = \\emptyset$.\n\\end{enumerate}\nExpress your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$\nare nonnegative integers.",
      "solution": "Solution. There is a bijection between triples of subsets of \\( \\{1, \\ldots, 10\\} \\) and \\( 10 \\times 3 \\) matrices with 0,1 entries, sending \\( \\left(voidgroup, emptycluster, nullfamily\\right) \\) to the matrix \\( scalarbody=\\left(bulkwhole\\right) \\) with \\( bulkwhole=1 \\) if \\( colmarker \\in voidgroup_{rowmarker} \\) and \\( bulkwhole=0 \\) otherwise. Under this bijection the set \\( singularone \\) of triples satisfying\n\\[\nvoidgroup \\cup emptycluster \\cup nullfamily=\\{1, \\ldots, 10\\} \\quad \\text { and } \\quad voidgroup \\cap emptycluster \\cap nullfamily=\\emptyset\n\\]\nmaps onto the set \\( monomatrix \\) of \\( 10 \\times 3 \\) matrices with 0,1 entries such that no row is (000) or (111). The number of possibilities for each row of such a matrix is \\( 2^{3}-2=6 \\), so \\( \\# monomatrix=6^{10} \\). Hence \\( \\# singularone=\\# monomatrix=2^{10} 3^{10} \\).\n\nReinterpretation. Equivalently, this problem asks for the number of ways of placing the numbers 1 through 10 in the Venn diagram of Figure 1, where no numbers are placed in the two regions marked with an \" \\( \\times \\) \".\n"
    },
    "garbled_string": {
      "map": {
        "A_1": "qzxwvtnp",
        "A_2": "hjgrksla",
        "A_3": "fzmbylqe",
        "B": "ksdjhqwe",
        "b_ij": "voskgmpl",
        "i": "zdtrqfwu",
        "j": "lsnkvhpe",
        "S": "xunopqra",
        "T": "ykrbsmte"
      },
      "question": "Determine, with proof, the number of ordered triples $(qzxwvtnp, hjgrksla, fzmbylqe)$ of sets which have the property that\n\\begin{enumerate}\n\\item[(i)] $qzxwvtnp \\cup hjgrksla \\cup fzmbylqe = \\{1,2,3,4,5,6,7,8,9,10\\}$, and\n\\item[(ii)] $qzxwvtnp \\cap hjgrksla \\cap fzmbylqe = \\emptyset$.\n\\end{enumerate}\nExpress your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.",
      "solution": "Solution. There is a bijection between triples of subsets of \\( \\{1, \\ldots, 10\\} \\) and \\( 10 \\times 3 \\) matrices with 0,1 entries, sending \\( \\left(qzxwvtnp, hjgrksla, fzmbylqe\\right) \\) to the matrix \\( ksdjhqwe=\\left(voskgmpl\\right) \\) with \\( voskgmpl=1 \\) if \\( zdtrqfwu \\in A_{lsnkvhpe} \\) and \\( voskgmpl=0 \\) otherwise. Under this bijection the set \\( xunopqra \\) of triples satisfying\n\\[\nqzxwvtnp \\cup hjgrksla \\cup fzmbylqe=\\{1, \\ldots, 10\\} \\quad \\text { and } \\quad qzxwvtnp \\cap hjgrksla \\cap fzmbylqe=\\emptyset\n\\]\nmaps onto the set \\( ykrbsmte \\) of \\( 10 \\times 3 \\) matrices with 0,1 entries such that no row is (000) or (111). The number of possibilities for each row of such a matrix is \\( 2^{3}-2=6 \\), so \\( \\# ykrbsmte=6^{10} \\). Hence \\( \\# xunopqra=\\# ykrbsmte=2^{10} 3^{10} \\).\n\nReinterpretation. Equivalently, this problem asks for the number of ways of placing the numbers 1 through 10 in the Venn diagram of Figure 1, where no numbers are placed in the two regions marked with an \" \\( \\times \\) \"."
    },
    "kernel_variant": {
      "question": "1. ENHANCED PROBLEM  \nLet Ω = {1,2,…,12}.  A 7-tuple (C₁,C₂,…,C₇) of subsets of Ω is called admissible when the following four conditions hold.  \n(i)  C₁∪C₂∪⋯∪C₇ = Ω  \n(ii) C₁∩C₂∩⋯∩C₇ = ∅  \n(iii) every element of Ω belongs to at least two of the seven sets  \n(iv) each of the seven sets has even size: |Cⱼ|≡0 (mod 2) for j = 1,…,7\n\nDetermine, with proof, the total number N of admissible 7-tuples and show that  \n\n  N = (119¹² + 28·3¹² + 28·5¹² + 7¹² + 70) / 2⁷.  \n\n(Observe that the right–hand side is an integer.)  ",
      "solution": "2. ENHANCED SOLUTION\n\nStep 1.  Encoding by incidence matrices  \nTo every ordered 7-tuple (C₁,…,C₇) we attach its 12×7 incidence matrix  \nM = (mᵢⱼ)  with mᵢⱼ = 1  if i∈Cⱼ and 0 otherwise.  \nThe map “tuple ↦ matrix’’ is a bijection.  \n\n• Conditions (i)–(iii) restrict the rows.  \n – (i) forbids the all-zero row;  \n – (ii) forbids the all-one row;  \n – (iii) forbids rows containing exactly one 1.  \n\n Consequently every row has Hamming weight 2,3,4,5 or 6.  The set  \n\n  S := { v∈𝔽₂⁷ | 2≤wt(v)≤6 }  \n\ncontains |S| = Σ_{k=2}^{6} C(7,k) = 119 vectors.  \n\n• Condition (iv) is a column condition:  \n for each column j the sum of the 12 entries m₁ⱼ+⋯+m₁₂ⱼ is even,  \n i.e. the 12 row vectors v₁,…,v₁₂ add up to 0 in the vector space V = 𝔽₂⁷:  \n\n  v₁+v₂+⋯+v₁₂ = 0.                                          (1)  \n\nThus N equals the number of 12-term sequences in S whose sum is 0.  \n\nStep 2.  A character sum (Fourier) evaluation  \nWrite χ_λ(v) = (−1)^{λ·v} for λ,v∈V and let  \n\n  σ(λ) = Σ_{v∈S} χ_λ(v)                            (2)  \n\nbe the Fourier transform of the characteristic function of S.  \nBy the standard “root-of-unity filter’’ (or orthogonality of characters),\n\n  N = 2^{−7} Σ_{λ∈V} σ(λ)^{12}.                       (3)  \n\nStep 3.  Computing σ(λ)  \nThe value σ(λ) depends only on r = wt(λ).  Put σ_r := σ(λ) for any λ of\nweight r (0≤r≤7).  Because V splits into r coordinates in the support of λ\nand 7−r outside the support, routine inclusion–exclusion gives\n\n  σ_r = Σ_{k=2}^{6} Σ_{s=0}^{k} (−1)^s C(r,s) C(7−r,k−s).        (4)\n\nA quicker way is to observe that Σ_{v∈V}χ_λ(v)=0 when λ≠0 and rewrite S as\nV minus the vectors of weight 0,1,7.  One obtains\n\n  σ_r = −[ 1 + (7−2r) + (−1)^r ] = 2r − 8 − (−1)^r.            (5)\n\nHence  \n\n σ₀ = 119, σ₁ = σ₂ = −5, σ₃ = σ₄ = −1, σ₅ = σ₆ = 3, σ₇ = 7.  \n\nStep 4.  Feeding σ_r into (3)  \nLet N_r = C(7,r) be the number of λ with wt(λ)=r.  Then\n\n N = 2^{−7} [ N₀σ₀^{12} + (N₁+N₂)(−5)^{12} + (N₃+N₄)(−1)^{12}  \n       + (N₅+N₆)3^{12} + N₇ 7^{12} ].\n\nInsert the binomial coefficients:\n\n N = 2^{−7} [ 1·119^{12} + 28·5^{12} + 70·1 + 28·3^{12} + 1·7^{12} ]  \n  = (119^{12} + 28·5^{12} + 28·3^{12} + 7^{12} + 70) / 2^{7}.   (6)\n\nThe numerator is even and in fact divisible by 2⁷, so the quotient is an\ninteger, completing the enumeration.",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.074756",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}