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Chapter 1: Q. 1.2 (page 15)

How many outcome sequences are possible when a die is rolled four times, where we say, for instance, that the outcome is 3, 4, 3, 1 if the first roll landed on 3, the second on 4, the third on 3, and the fourth on 1?

Short Answer

Expert verified

The number of possible sequences are6脳6脳6脳6=64

Step by step solution

01

  Step 1 .Given information

Here a die is rolled four times, we have to find out the possible number of sequences of out comes

02

.  Finding the  number of possible sequences of outcomes

If a die is rolled for on time, the outcomes are 1,2,3,4,5,6, that is 6 outcomes. Then the die is rolled four times, the outcomes are repeated four timesrole="math" localid="1647497151106" thatis6脳6脳6脳6=64times

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Most popular questions from this chapter

An art collection on auction consisted of 4Dalis, 5van Goghs, and 6Picassos. At the auction were 5 art collectors. If a reporter noted only the number of Dalis, van Goghs, and Picassos acquired by each collector, how many different results could have been recorded if all of the works were sold?

Consider the following combinatorial identity:

鈭�k=1nknk=n路2n-1

(a) Present a combinatorial argument for this identity by considering a set of npeople and determining, in two ways,

the number of possible selections of a committee of any size and a chairperson for the committee.

Hint:

(i) How many possible selections are there of a committee of size kand its chairperson?

(ii) How many possible selections are there of a chairperson and the other committee members?

(b) Verify the following identity for n=1,2,3,4,5:

localid="1648098528048" 鈭�k=1nnkk2=2n-2n(n+1)

For a combinatorial proof of the preceding, consider a set of n people and argue that both sides of the identity represent

the number of different selections of a committee, its chairperson, and its secretary (possibly the same as the chairperson).

Hint:

(i) How many different selections result in the committee containing exactly kpeople?

(ii) How many different selections are there in which the chairperson and the secretary are the same?

(answer: n2n鈭�1.)

(iii) How many different selections result in the chairperson and the secretary being different?

(c) Now argue that

localid="1647960575612" 鈭�k=1nnkk3=2n-3n2(n+3)

Prove that:

n+mr=n0mr+n1mr-1+..........+nrm0

Hint: Consider a group of nmen and mwomen. How many groups of size rare possible?

Give a combinatorial explanation of the identity

nr=nn-r

Give an analytic verification of

n2=k2+k(n-k)+n-k2,1鈮�k鈮�n

Now, give a combinatorial argument for this identity.
See all solutions

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