Challenging Mathematical Problems with Elementary Solutions: Combinatorial analysis and probability theoryHolden-Day, 1964 |
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第 1 到 3 筆結果,共 55 筆
第 9 頁
... Prove that if n > m ( m + 1 ) / 2 , the number of partitions of n into m distinct parts is equal to the number of partitions of n m ( m + 1 ) / 2 into at most m ( not necessarily distinct ) parts . 33a . * Prove that the number of ...
... Prove that if n > m ( m + 1 ) / 2 , the number of partitions of n into m distinct parts is equal to the number of partitions of n m ( m + 1 ) / 2 into at most m ( not necessarily distinct ) parts . 33a . * Prove that the number of ...
第 199 頁
... prove that for any positive integer M there exists an n such that 2 " begins with the sequence of digits which represents the number M in the decimal notation . This is equivalent to proving that for any positive integer M one can find ...
... prove that for any positive integer M there exists an n such that 2 " begins with the sequence of digits which represents the number M in the decimal notation . This is equivalent to proving that for any positive integer M one can find ...
第 230 頁
... prove this fact one needs only to show that Sn = Gn - 1 ( compare with the solution to problem 54 ) . 85a . 1 • m + n b . Show that the desired probability P is independent of k . From 1 this it follows that P = k m + n 86. 1/3 ; 5/9 ...
... prove this fact one needs only to show that Sn = Gn - 1 ( compare with the solution to problem 54 ) . 85a . 1 • m + n b . Show that the desired probability P is independent of k . From 1 this it follows that P = k m + n 86. 1/3 ; 5/9 ...
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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