CSCI2110 Tutorial 6:More Counting by Mapping, Number Sequences

Chow Chi Wang (cwchow ‘at’ cse.cuhk.edu.hk)

More Counting by Mapping

Division Rule• If a function f from A to B is k-to-1 (this means for every

element in B is mapped by exactly k elements in A.), then

.

• Sometimes we can’t find a direct way to count the size of a set A.

• The idea of the division rule is to establish a k-to-1 correspondence between A and another set B, which is hopefully more easily countable.

Division Rule (Example)• (1) Consider the string NILLAPALOO in this question.

a) How many distinguishable ways can the letters be arranged in order?

• Let A be the set of all possible rearrangement of the string.

• Let B be the set of the strings of length 10 constructed by these symbols S = {N, I, L1, L2, L3, A1, A2, P, O1, O2}.

• Consider the function , where f replaces every symbols in S by the corresponding letter.• e.g. f(NIL1L2L3A1A2PO1O2) = NILLLAAPOO

Division Rule (Example)• We know that:

• Because there are 10 choices for the 1st component of the string• There are 9 choices for the 2nd component of the string• …• So in total there are 10x9x…x1 = 10! many possible strings in B.

• f is a -to-1function.• Strings of the forms and map to the same string in A by f. And there are 2!

such forms corresponds to ’s.• Similarly for the ’s and ’s, which have 3! and 2! different forms to consider

respectively.

• So by the division rule, we know that:

Number Sequences

Sum of a Sequence (Series)• (Sigma Notation) .

• Some common facts:• .• .• .

Telescoping Series• A telescoping series is a series whose partial sums

eventually only have a fixed number of terms after cancellation.

• e.g.

Telescoping Series (Puzzle)• 0 = 1?

• Consider:

!

• What is wrong?

Arithmetic Sequence• An arithmetic sequence is a sequence of numbers such

that the difference between the consecutive terms is constant

• e.g. 1, 3, 5, 7, 9, …

• In general, arithmetic sequence can be expressed as:• (initial value)• for ( is called the common difference)

Arithmetic Series• Let and is an arithmetic sequence.

• We have:

• Because is an arithmetic sequence, we have:

Arithmetic Series• Because there are such terms so:

Therefore:

Geometric Sequence• A geometric sequence is a sequence of numbers where

each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio.

• e.g. 1, 2, 4, 8, 16, …

• In general, arithmetic sequence can be expressed as:• (initial value)• for ( is called the common ratio)

Geometric Series• Let and is a geometric sequence.

• We have:

• Therefore, for :

Harmonic Series• We define to be the n-th harmonic number.

• Harmonic series is divergent! This means

• This is because we have:

•

• So we have , therefore harmonic series is divergent.

Harmonic Series• Here is a bound for the harmonic numbers:

• We can prove this by integration.

Pi Notation• (Pi Notation) .

Thank You!