COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE

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145 REVISED 62016 ACCOUNTING PRINCIPLES AND STANDARDS HANDBOOK CHAPTER
15 COMMENTS DUTCH ACCOUNTING STANDARDS BOARD (DASB) ON EXPOSURE
2 APPENDIX B RULES FOR COUNTING THE NUMBER OF

Teknik Pengiraan

Counting Technique

COUNTING TECHNIQUES


OBJECTIVES

  1. Student should be able to understand all types of counting techniques.

  2. Students should be able to identify the three techniques learned.

  3. Students should be able to use each of the counting techniques based on different questions and situations.


What, Which, Where, When

  1. Permutation

Redundant elements (Clear / Not Clear)

Non-redundant elements (Clear / Not Clear)

The use of nPr (Clear / Not Clear)

  1. Combination

Non-redundant elements (Clear / Not Clear)

The use of nCr (Clear / Not Clear)

  1. Pigeonhole

Pigeonhole Principle (Clear / Not Clear)

Identifying n and m (Clear / Not Clear)


COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE

Kolman, Busby and Ross page 78 – 91

Rosen 4th Ed page 232 – 259

Rosen 5th Ed page 301 – 326

Jonsonbaugh page 165 – 218

Mattson page -

COUNTING TECHNIQUES


<, how many words that can be build with length 3, repetition allowed?

n = 4, r = 3, then nr = 43 = 64 words


A sequence of r elements from n elements of A is always said as ‘permutation of r elements chosen from n elements of A’, and written as nPr or P(n, r)


If 1 r n, then nPr is the number of permutation of n objects taken r at a time, is

n(n-1)(n-2)… (n-(r - 1))


When r = n, that is from n objects, taken r at a time from A, where r = n, it is a nPn or n factorial, written as n!.


<

COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE 4P3 = 4.3.2.1

1

= 4.3.2

= 24 (ex: pqr, pqs, prq, prs, psq, psr, …….)

Ex 5: Choose 3 alphabets from A..Z

COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE 26P3 = 26.25.24.23 …. 3.2.1

23.22……3.2.1.

= 26.25.24


The number of distinguishable permutations that can be formed from a collection of n objects where the first object appears k1 times, the second object appears k2 times, and so on, is:

COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE n!

k1!k2!…ki!

Ex 6: a) MISSISSIPPI b) CANADA


Combination

Let A be a set with |A| = n, and let 1 r n. Then the number of combinations of the elements of A, taken r at a time, written as nCr, is given by

COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE nCr = n!

r! (n - r)!

Ex 7: If A = {p, q, r, s}, find the number of combination for 3 elements.

COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE 4C3 = 4.3.2.1

3.2.1.1

= 4 (ex: pqr, pqs, prs, qrs) (pqr, prq, rpq, rqp, all are the same)


Suppose k selections are to be made from n items without regard to order and repeats are allowed, assuming at least k copies of each of the n items. The number of ways these selections can be made is (n + k -1)Ck.

Ex 8: In how many ways can a prize winner choose three CDs from the Top Ten list if repetition is allowed?

COUNTING TECHNIQUE COUNTING TECHNIQUES OBJECTIVES 1 STUDENT SHOULD BE n = 10 and k = 3, so, (10 + 3 -1)C3 = 12C3 = 12.11.10.9.8….1

3.2.1.9.8….1

= 2.11.10

= 220 ways.

Pigeonhole

If there are n pigeon are assigned to m pigeonhole, where m < n, then at least one pigeonhole contains two or more pigeons.

Ex 9: if 8 people were chosen, at least 2 people were being born in the same day (Monday to Sunday). Show that by using pigeonhole principle.

Because there are 8 people and only 7 days per week, so Pigeonhole Principle says that, at least two or more people were being born in the same day.



Ex 10: Show that if any five numbers from 1 to 8 are chosen, two of then will add to 9.

Two numbers that add up to 9 are placed in sets as follows:

A1 = {1, 8}, A2 = {2, 7}, A3 = {3, 6}, A4 = {4, 5}

Each of the 5 numbers chosen must belong to one of these sets. Since there are only four sets, the pigeonhole principle tells us that two of the chosen numbers belong to the same set. These numbers add up to 9.


If there are m pigeonholes and more than 2m pigeons, three or more pigeons will have to be assigned to at least one of the pigeonholes.

If n and m are positive integers, then n/m stands for largest integer less than equal to the rational number n/m.

3/2 = 1, 9/4 = 2 6/3 = 2


If n pigeons are assigned to m pigeonholes, then one of the pigeonholes must contain at least

(n-1)/m + 1 pigeons.


Exercise:

  1. Find the number of order to choose 3 letters from the word COMPUTER

  1. Without repetition

  2. With repetition

  1. How many choice are there if the student must answer:

  1. 8 questions out of 10 questions?

  2. 8 questions out of 10, but the first 3 are compulsory questions.

  1. Find the number of choice to choose 3 men and 5 women from a group of 23 men and 14 women.

  2. Given ABCDEF as 6 persons. How many ways are there to make sure that DEF always sits next to each other, in that order.

  3. Given ABCDEF as 6 persons. How many ways are there to make sure that DEF always sits next to each other but not necessarily in that order.

  4. Show that if there are 30 students in a class, at least the name of 2 students must start with the same letter.

  5. How many students should be in a class to ensure that at least 5 students get the same grade if grades available are A, B C, D and E?

6



20 41 ACCOUNTING FOR PENSIONS AND POSTRETIREMENT BENEFITS
23 FEBRUARY 2006 SIR DAVID TWEEDIE CHAIRMAN INTERNATIONAL ACCOUNTING
24 JUNE 2009 SIR DAVID TWEEDIE INTERNATIONAL ACCOUNTING STANDARDS


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