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Q. 1.10

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A First Course in Probability
Found in: Page 17
A First Course in Probability

A First Course in Probability

Book edition 9th
Author(s) Sheldon M. Ross
Pages 432 pages
ISBN 9780321794772

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Short Answer

From a group of n people, suppose that we want to choose a committee of k, k≤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" nkk possible 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+1 possible 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-1 possible 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 of mr to verify the identity in part (d).

(a) The possible number of choices are nkk

(b) The possible number of choices are nk-1n-k+1

(c) The possible number of choices are nn-1k-1

(d) It is proved that knk=n-k+1nk-1=nn-1k-1

(e) It is proved that knk=n-k+1nk-1=nn-1k-1

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Part (a) Step 1. Find the possible number of choices.

Total no. of persons =n

No. of persons to be chosen for committee =k

So, the number of members for the committee can be chosen in localid="1647947695401" Ckn ways = nkways

Out of k members, one can be selected as chairperson in C1k ways = k ways

Therefore, total no. of ways are nkk.

Part (b) Step 1. Find the possible number of choices.

Out of k members, 1 member will be the chair person. So, the non-chair members will be k-1.

So, non-chair members can be selected out of n members in role="math" localid="1647948253033" Ck-1nways =nk-1ways.

The left number of persons will be role="math" localid="1647948086924" n-(k-1)=n-k+1

Out of the left members, one can be selected as chairperson in C1n-k+1 ways =n-k+1 ways.

Therefore, total no. of ways are nk-1n-k+1.

Part (c) Step 1. Find the possible number of choices.

Out of n persons, one can be selected as chairperson in role="math" localid="1647948578979" C1n ways =n ways.

The remaining no. of people will be =n-1

The number of non-chair members of the committee will be =k-1

So, non-chair members can be selected out of n-1 members in Ck-1n-1ways = n-1k-1ways

Therefore, total no. of ways are nn-1k-1.

Part (d) Step 1. Give conclusion.

In all the above parts, (a), (b) and (c), we have found the ways in which we could select k committee members out of the group of n persons and one chair person of the committee.

Therefore, the total number of ways will be equal.

Hence, it is proved that role="math" localid="1647950480646" knk=n-k+1nk-1=nn-1k-1

Part (e) Step 1. Verify the identity.

As per the factorial definition, mr=m!r!(m-r)!.

Therefore,

knk=k×n!k!(n-k)!=k×n!k×(k-1)!(n-k)!=n!(k-1)!(n-k)!................ (1)

n-k+1nk-1=n-k+1×n!(k-1)!(n-k+1)!=n-k+1×n!(k-1)!(n-k+1)×(n-k)!=n!(k-1)!(n-k)!............ (2)

nn-1k-1=n×(n-1)!(k-1)!(n-1-k+1)!=n×(n-1)!(k-1)!(n-k)!=n!(k-1)!(n-k)!............... (3)

As (1) = (2) = (3)

Therefore, it is proved that knk=n-k+1nk-1=nn-1k-1

Most popular questions for Math Textbooks

Expandx1+2x2+3x34.

Let Hk(n) be the number of vectors x1, . . . , xk for which each xi is a positive integer satisfying 1≤ xi≤n and x1 ≤ x2≤, … , ≤xk.

(a)Without any computations, argue that

localid="1648218400232" H1(n)=nHk(n)=∑j=1nHk-1(j) k>1

Hint: How many vectors are there in which xk = j?

(b) Use the preceding recursion to compute H3(5).

Hint: First compute H2(n) for n = 1, 2, 3, 4, 5.

Consider a tournament of n contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let localid="1648231792067" N(n) denote the number of different possible outcomes. For instance, localid="1648231796484" N(2) = 3, since, in a tournament with localid="1648231802600" 2 contestants, player localid="1648231807229" 1 could be uniquely first, player localid="1648231812796" 2 could be uniquely first, or they could tie for first.

(a) List all the possible outcomes when n=3.

(b) With localid="1648231819245" N(0) defined to equal localid="1648231826690" 1, argue without any computations, that localid="1648281124813" N(n)=∑i=1nniNn-i

Hint: How many outcomes are there in which localid="1648231837145" i players tie for last place?

(c) Show that the formula of part (b) is equivalent to the following:

localid="1648285265701" N(n)=∑i=1n-1niNi

(d) Use the recursion to find N(3) and N(4).

For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and 9, the second digit was either 0 or 1, and the third digit was any integer from 1 to 9. How many area codes were possible? How many area codes starting with a 4 were possible

A student is to answer 7 out of 10 questions in an examination. How many choices has she? How many if she must answer at least 3 of the first 5 questions?
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