Challenging Mathematical Problems with Elementary Solutions, 第 1 卷Holden-Day, 1964 - 440 頁 |
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第 52 頁
... expression # ( A ) + # ( B ) counts the number of elements in A or B , but counts the elements of AB twice . So by subtracting # ( A ◇ B ) we get exactly the number of elements in AU B. 12b . Consider an element e which is in only one ...
... expression # ( A ) + # ( B ) counts the number of elements in A or B , but counts the elements of AB twice . So by subtracting # ( A ◇ B ) we get exactly the number of elements in AU B. 12b . Consider an element e which is in only one ...
第 126 頁
... expression to be evaluated reduces to the single term 57f . The required sum is the coefficient of x " in the polynomial x " ( 1 − x ) " + x ” −1 ( 1 − x ) " + ··· + xn − m ( 1 − x ) " . - Transforming this polynomial , we obtain ...
... expression to be evaluated reduces to the single term 57f . The required sum is the coefficient of x " in the polynomial x " ( 1 − x ) " + x ” −1 ( 1 − x ) " + ··· + xn − m ( 1 − x ) " . - Transforming this polynomial , we obtain ...
第 195 頁
... expression approaches the limit 2/6 = 1/3 . Therefore the probability that a positive integer selected at random is relatively prime to 6 equals 1/3 . The probability that at least one of two integers selected at random from the ...
... expression approaches the limit 2/6 = 1/3 . Therefore the probability that a positive integer selected at random is relatively prime to 6 equals 1/3 . The probability that at least one of two integers selected at random from the ...
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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 given Hence inclusion and exclusion intersection k-gons knights L₁ length mathematical induction maximum number n-gon number of different number of favorable number of paths number of shortest obtain pairs partition passengers plane points A1 polygons positive integers possible outcomes Pr{E probability theory problem 54 prove queens rectangle relatively prime remaining required probability rooks segment selected at random sequence shortest paths side solution to problem solved sphere square controlled Suppose T₂ total number triangle unfavorable values vertex vertices