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Chapter 1: Q. 1.15 (page 20)

A total of nstudents are enrolled in a review course for the actuarial examination in probability. The posted

results of the examination will list the names of those who passed, in decreasing order of their scores. For instance, the posted result will be 鈥淏rown, Cho鈥� if Brown and Cho are the only ones to pass, with Brown receiving the higher score. Assuming that all scores are distinct (no ties), how many posted results are possible?

Short Answer

Expert verified

The possible no. of posted results are 鈭�k=0nnkk!

Step by step solution

01

Step 1. Given information.

It is given that,

Total no. of students enrolled in a review course for the actuarial examination =n.

The posted results of the examination will list the names of those who passed, in decreasing order of their scores.

02

Step 2. Find the possible no. of posted results.

If kpeople pass then there are nkdifferent groups of size k.

Further there are k!possible ordering of their scores.

Therefore, there are nkk!possible results in which kpeople pass.

The possible no. of posted results are鈭�k=0nnkk!.

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Most popular questions from this chapter

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)

Suppose that 10fish are caught at a lake that contains 5distinct types of fish.

(a)How many different outcomes are possible, where an outcome specifies the numbers of caught fish of each of the 5types?

(b)How many outcomes are possible when3the 10fish caught are trout?

(c)How many when at least 2of the 10are trout?

From a group of npeople, suppose that we want to choose a committee of k, k鈮�n, one of whom is to be designated as chairperson.

(a) By focusing first on the choice of the committee and then on the choice of the chair, argue that there are role="math" localid="1647945358534" nkkpossible choices.

(b) By focusing first on the choice of the non-chair committee members and then on the choice of the chair, argue that there are role="math" localid="1647945372759" nk-1n-k+1possible choices.

(c) By focusing first on the choice of the chair and then on the choice of the other committee members, argue that

there are role="math" localid="1647945385288" nn-1k-1possible choices.

(d) Conclude from parts (a), (b), and (c) that role="math" localid="1647945400273" knk=n-k+1nk-1=nn-1k-1.

(e) Use the factorial definition ofmr to verify the identity in part (d).

An elevator starts at the basement with 8people (not including the elevator operator) and discharges them all by the time it reaches the top floor, number6. In how many ways could the operator have perceived the people leaving the elevator if all people look alike to him? What if the 8people consisted of 5men and 3women and the operator could tell a man from a woman?

Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to selected students

from a class of 30. How many different outcomes are possible if

(a) a student can receive any number of awards?

(b) each student can receive at most 1award?
See all solutions

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