Challenging Mathematical Problems with Elementary Solutions: Combinatorial analysis and probability theoryHolden-Day, 1964 |
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第 46 頁
... shown in fig . 24b , and the other three are each inscribed in an interior dihedral angle and the opposite exterior ... shown in figs . 24a and 24b ) . Remark . Let A1 , A2 , A3 , A4 be the areas of the four faces of the tetrahedron T ...
... shown in fig . 24b , and the other three are each inscribed in an interior dihedral angle and the opposite exterior ... shown in figs . 24a and 24b ) . Remark . Let A1 , A2 , A3 , A4 be the areas of the four faces of the tetrahedron T ...
第 72 頁
... shown in fig . 26. Any solution of the equation x1 + ··· + xm = n in positive integers corresponds to a decomposition of this segment into m pieces whose lengths are positive integers . The m - 1 end points of these pieces ( other than ...
... shown in fig . 26. Any solution of the equation x1 + ··· + xm = n in positive integers corresponds to a decomposition of this segment into m pieces whose lengths are positive integers . The m - 1 end points of these pieces ( other than ...
第 73 頁
... shown in fig . 27a ( where m = 4 ) . There are 1 + 2 + 3 + ··· + m = m ( m + 1 ) / 2 dots in this triangle . Suppose now that this triangle is removed , and that the remaining dots are then shifted to the left as shown in fig . 27b ...
... shown in fig . 27a ( where m = 4 ) . There are 1 + 2 + 3 + ··· + m = m ( m + 1 ) / 2 dots in this triangle . Suppose now that this triangle is removed , and that the remaining dots are then shifted to the left as shown in fig . 27b ...
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A₁ A₂ An+m arrangements b₁ B₂ binomial coefficients binomial theorem bishops black squares C₁ chessboard chord circle coefficient color column compute the number Consequently consider corresponding customers denote determine the number diagonals digits dihedral angle divided divisible draw equally likely possible equation equidistant equivalence classes exactly example experiment favorable outcomes follows formula G₁ given Hence inclusion and exclusion intersection k-gons knights length mathematical induction maximum number n-gon number of different number of favorable number of paths number of shortest obtain pairs partition passengers plane polygons positive integers possible outcomes Pr{E probability theory problem 54 prove queens rectangle relatively prime remaining required probability rooks S₁ segment selected at random sequence shortest paths side solution to problem solved sphere square controlled Suppose T₂ total number triangle unfavorable values vertex vertices