A%20Probability%20Approach%20For%20Solving%20Counting%20Problems

A%20Probability%20Approach%20For%20Solving%20Counting%20Problems

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1 A Probability Approach for Solving Counting Problems There are some GMAT questions where solving is not the problem! The problem is time! Let me give a simple example: How many times will the digit 7 be written when listing the integers from 1 to 1000? A) 110 B) 111 C) 271 D) 300 E) 304 This is an easy question to solve. All you need in order to solve this question is a pen, a paper and time. On the GMAT you have a pen and a paper (actually you don't - you have a wipe off board and an erase marker, but that's another problem) but you don't have time. In order to solve this problem in the most basic possible way you simply need to write down the numbers from 1 to 1000 and just count the sevens – using an Excel spreadsheet it took me more then two minutes which is the allocated time per question on the quantitative section of the GMAT. (The detailed solution will be discussed later in this paper). I have seen some very good shortcuts for solving this question (and others like it). But I would like to offer you a fast and foolproof method. My method is probability based and I call it "A probability approach for solving counting problems" method. Now let Y be the ... don't worry! No Y is needed, just wanted to sound smart! (And as you know every good math/physics/finance paper starts with "let something be something") So I would like to start with a simple problem, very simple problem: How many integers from 1 to 100 including, has a units digit of 5? A) 5 B) 9 C) 10 D) 11 E) 20 This is a 350 level question; all you need to do is to list the integers that has 5 as the units digit, I call this the basic method, meaning Count(5,15,25,35,45,55,65,75,85,95) = 10. The answer is (C). Using the probability method we can see we have two places for digit in the integer XX and we need to fill in the tens digit and units digit (i.e. 33 or 51 or 03) each of them represent a number from 1 to 99 (for 01 = 1).
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