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Chapter 2: Q.2.17 (page 49)

If 8 rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

Short Answer

Expert verified

The probability that no row or file contains more than one rook9.109脳10-6

Step by step solution

01

Step1

Since rooks attack horizontally & vertically, you can't have a rook in the same row or column as another.

So in the first row, you place a rook. There are 8possible places for first one. In the next row you place a rook,it can't be in the same column as other rook so for this one we have 7possibilities i.e. 8!ways

Now total number of possible ways is648

so the probability that no rooks capturing =6488!=9.109脳10-6

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Most popular questions from this chapter

An instructor gives her class a set of 10problems with the information that the final exam will consist of a random selection of 5them. If a student has figured out how to do 7the problems, what is the probability that he or she will answer correctly

(a)all 5problems?

(b) at least4of the problems?

A woman has n keys, of which one will open her door.

(a) If she tries the keys at random, discarding those that do not work, what is the probability that she will open the door on her kth try?

(b) What if she does not discard previously tried keys?

Consider the matching problem, Example5m, and define it to be the number of ways in which theNmen can select their hats so that no man selects his own.

Argue thatAN=(N鈭�1)(AN-1+AN-2). This formula, along with the boundary conditionsA1=0,A2=1, can then be solved forAN, and the desired probability of no matches would be AN/N!.

Hint: After the first man selects a hat that is not his own, there remain N鈭�1men to select among a set of N鈭�1hats that do not contain the hat of one of these men. Thus, there is one extra man and one extra hat. Argue that we can get no matches either with the extra man selecting the extra hat or with the extra man not selecting the extra hat.

Five people, designated as A,B,C,D,E, are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that

(a) there is exactly one person between Aand B?

(b) there are exactly two people between Aand B?

(c) there are three people between AandB?

If there are 12 strangers in a room, what is the probability that no two of them celebrate their birthday in the same month?
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