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Chapter 1: Q. 1.5 (page 19)

In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each?

Short Answer

Expert verified

The no. of ways are 210.

Step by step solution

01

Step 1. Given information.

It is given that,

Total no. of gifts are =7

Total no. of children =3

Gifts to be given to eldest child=3

Gifts to be given to other two children=2each

02

Step 2. Find the no. of ways.

The no. of ways in which the eldest child will receive 3gifts and other two children will receive 2gifts each out of 7gifts is

73,2,2=7!3!2!2!=7×6×5×4×3!2×2×3!=7×6×5=210

Therefore, the no. of ways are210.

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Most popular questions from this chapter

In how many ways can 3novels, 2mathematics books, and 1chemistry book be arranged on a bookshelf if

(a) the books can be arranged in any order?

(b) the mathematics books must be together and the novels must be together?

(c) the novels must be together, but the other books can be arranged in any order?

Consider a tournament of ncontestants 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" 2contestants, player localid="1648231807229" 1could be uniquely first, player localid="1648231812796" 2could 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" iplayers 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).

From a set of npeople, a committee of size jis to be chosen, and from this committee, a subcommittee of size i, i≤j, is also to be chosen.

(a) Derive a combinatorial identity by computing, in two ways, the number of possible choices of the committee and subcommittee—first by supposing that the committee is chosen first and then the subcommittee is chosen, and second

by supposing that the subcommittee is chosen first and then the remaining members of the committee are chosen.

(b) Use part (a) to prove the following combinatorial identity:role="math" localid="1648189818817" ∑j=innjji=ni2n-i;i≤n

(c) Use part (a) and Theoretical Exercise 13 to show that:role="math" localid="1648189841030" ∑j=innjji-1n-j=0;i<n

A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. If the committee must consist of at least 3 women and at least 2 men, how many different committees are possible?

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
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