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 26 筆
第 58 頁
... occurs . ( 1 ) Exactly one factorization , namely occurs only once . 106 - ( 22.52 ) ( 22 · 52 ) ( 22 · 52 ) , ( 2 ) If in a decomposition of 106 into a product of three factors , two of the factors are equal ( and the third factor ...
... occurs . ( 1 ) Exactly one factorization , namely occurs only once . 106 - ( 22.52 ) ( 22 · 52 ) ( 22 · 52 ) , ( 2 ) If in a decomposition of 106 into a product of three factors , two of the factors are equal ( and the third factor ...
第 74 頁
... occurs in this partition , ... 9 Sk - 1 the number of times k the number of times k + 1 occurs , etc .: - 1 occurs , Sk + 1 n = s1 · 1 + ··· + Sx - 1 ( k − 1 ) + Sx + 1 ( k + 1 ) + ··· • Now write each of the numbers S1 , . . . , Sk ...
... occurs in this partition , ... 9 Sk - 1 the number of times k the number of times k + 1 occurs , etc .: - 1 occurs , Sk + 1 n = s1 · 1 + ··· + Sx - 1 ( k − 1 ) + Sx + 1 ( k + 1 ) + ··· • Now write each of the numbers S1 , . . . , Sk ...
第 75 頁
... occurs more than k 1 times , one can recover the original partition of n into parts not divisible by k . It follows from this that for any n there are as many partitions of the one kind as of the other . Second solution . We present ...
... occurs more than k 1 times , one can recover the original partition of n into parts not divisible by k . It follows from this that for any n there are as many partitions of the one kind as of the other . Second solution . We present ...
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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