Combinatorics by N. Ya. Vilenkin (Auth.)

By N. Ya. Vilenkin (Auth.)

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Extra resources for Combinatorics

Example text

This proves the relation n n «i = £rA,. r = 2>c;U)„_r. r=0 (5') r=0 The principle of inclusion and exclusion enables us to solve the following problem: Find the number of permutations of n elements in which r prescribed elements are displaced (and the remaining are either displaced or remain fixed). The required number is given by the formula »! - Cl(n - 1)! + C\n - 2)! - - + ( - 1 ) > - r)!. (6) Subfactorials* The numbers Dn are sometimes referred to as subfactorials. These numbers share many properties with ordinary factorials.

What counts is the average number of wins computed over a long period of time for a large number of players. Failure to understand this point may result in a faux pas of the kind attributed to a doctor who said to his patient: "Your illness is fatal in 9 out of 10 cases. The last 9 of my patients who suffered from your illness died. " Now let us compute the chances of winning when the ticket has 2 numbers. In this case 3 of the 5 numbers on the tokens selected from the bag are arbitrary. 30 II.

These problems can be formulated as follows: There are n different types of objects. We are required to compute the number of ^-arrangements of these objects without regard to order (in other words, two arrangements are different only if they differ in the number of elements of at least one type). The general problem is solved very much like the problem about pastries. Namely, each arrangement is coded by means of one and zeros, with each type represented by as many ones as there are elements of this type in the arrangement and with different types separated by zeros (if objects of certain types are absent then we write two or more zeros in a row).

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