Sunday, September 1, 2013

Counting & Venn Diagram Part: Counting, Making a Systematic list

Counting & Venn diagram:
In some questions you are asked to count something: How many apples did Jorge buy, how many taka did Shelly spend, how many pages were read,  how many ways are there to complete a job, etc. Some such problems may be solved using algebra. Others may need your to systematically make a list. Sometimes you may need to know some special strategies that we discussed.

Making a systematic list: 
When a question asks 'how many', often the best strategy is to make a list of all the possibilities. You have to make this list systematically so that no item is left out or so that you don't count an item twice. Usually this means listing the possibilities in a numerical or alphabetical order. Inmost case, shortly, After starting to make the systematic list, you world see a pattern developing and
 you should be able to find out how many elements should be in the list without having to write drown all of them.

Example: A palindrome is a number, such a 93539, that reads the same forward and backward. How many palindromes are there between 100 and 1000?

Solution: At first let us write the palindromes between 100 and 200.
101, 111, 121,131,141,151, 161, 171, 181, 191 = 10 Palindromes
Next comes the number between 200 and 300.
202, 212, 222, 232, 242, 252, 262, 272, 282, 292 =10 Palindromes
So, we can see a pattern. Each interval of 100 numbers contains 10 Palindromes. So, beginning with 1, 2, 3,.. 9 there are 9*10 = 90 Palindromes.
Answer: 90

This a good technique if the numbers in the problem are small. Proper use of this technique eliminates the risk of marking an error in arithmetic.

Using Arithmetic to count: 
Let us go through a few examples. These will illustrate the basic rules of arithmetic we use  for counting..
Example A:  Bari entered the store with $213 & left with $ 203. How much did he spend in the store?

Example B: A man sitting in a cinema hall ticket counter sold ticket no. 213 through 209. How many tickets were sold?

Example C: Bari is the 213th person in a queue and Shaki is the 209th person. How many people are there between them?

Solution:  (A-c): At first sight it may seem that all the three have the same answer 213-209=4. But actually, the answer for A is 4. B is 5 and C is 3.
In example A, Bari had $ 213 He spent $ no. 213, 212, 211, 210. He still had $ 2.9 left with him. So, he spend $4.
In example B. tickets numbered 213, 212, 211, 210, & 209 were sold . So, the total number is 5.
 In the queue (example C) between Bari & Shakily there are person no. 212, 211 & 210 . So, the answer is 3.

In example A, we just subtracted, in example B we subtracted & then added a 1, in example C we subtracted & then subtracted 1 more.

The main point to note is whether or not the first and last numbers are included. In example A, Bari left the store with the $ 209. In example B. both the 213 & 209 ticket were sold. And in example C. neither the 213 person nor the 209 person was counted. So, ultimately we come to the following rules:

A) If exactly one of the end points is included, subtract.
B) If both the end points are included, subtract and add1 .
C) If none of the end points are included, subtract & subtract 1 more.

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