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Chapter 1: Q. 1.13 (page 18)

Show that, for n > 0,

∑i=0n(-1)ini=0

Hint: Use the binomial theorem.

Short Answer

Expert verified

It is proved that, for n>0

localid="1648188854365" ∑i=0n(-1)ini=0

Step by step solution

01

Step 1. Given information.

Here, we have to prove that, for n>0

02

Step 2. State the Binomial Theorem

According to the Binomial Theorem

(x+y)n=∑i=0nnixiyn-i

03

Step 3. Prove that ∑i=0n(-1)ini=0

Let us assume x=-1andy=1in (x+y)n=∑i=0nnixiyn-i

(-1+1)n=∑i=0nni-1i1n-i⇒0n=∑i=0nni-1i×1⇒0=∑i=0nni-1i

Therefore, it is proved that∑i=0n(-1)ini=0

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Most popular questions from this chapter

Consider n-digit numbers where each digit is one of the 10integers 0,1,...,9. How many such numbers are there for which

(a) no two consecutive digits are equal?

(b) 0 appears as a digit a total of itimes, i=0,...,n?

A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from a group of 5 Republicans, 6 Democrats, and 4 Independents. How many committees are possible?

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?

Prove the generalized version of the basic counting principle.

Consider the following combinatorial identity:

∑k=1nknk=n·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" ∑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−1.)

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

(c) Now argue that

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