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What does the word “to ensure” in combinatorical problems mean?


polyhedraldiceOn the test you can see some problems which ask you to find minimum number of attempts “to ensure” (or “to guarantee”) some outcome. These tasks invite us to provide a foolproof solution that would work in 100% of the cases.  To solve such problems, you need to remember one simple rule: there are no formulas, just count the worst scenario.  

Example 1.


A drugstore has 80 different kinds of drugs on its easy access shelves: 30% are for allergy, 25% are laxative, 20% are for cold, 5% are for headaches, and the rest are pain relief medicines. A blind person walks into the store to get some medicine for her cold. Assuming there are 20 bottles of each individual drug on the shelves and the blind customer does not ask for assistance, how many bottles will she need to purchase to ensure she has at least one bottle of cold medicine?

 

(A) 61
(B) 120
(C) 800
(D) 1281
(E) 1321


As we can see the word “to ensure” in the text, this is the so - called worst scenario problem. So, we need to pick the absolutely worst scenario to get the proper answer. Let’s assume, the blind customer buys all the other drugs before reaching to the Cold aisle. There are 1,600 different drugs and 20% or 320 are cold relief. Thus 1600 - 320 = 1280 are non cold. Therefore, she needs to buy all these non-suitable bottles and then 1 more, which will be from cold with the probability of 100%. So, she has to buy 1281 bottles of pills to get at least one needed medicine kind, assuming the worst scenario. The right answer is D.


Example 2.


Of the science books in a certain supply room, 50 are on botany, 65 are on zoology, 90 are on physics. 50 are on geology, and 110 are on chemistry. If science books are removed randomly from the supply room, how many must be removed to ensure that 80 of the books removed are on the same science?

 

(A) 81
(B) 159
(C) 166
(D) 285
(E) 324

The word “to ensure” in the task this means we will need to find a solution for the worst case. So after we will have removed all of the botany and geology books as well as 65 on zoology, 50 on botany, and 50 on geology, but we still don't have 80 of the same kind. So, after another 79 on chemistry and 79 on physics still not enough. Now we will have a total of 50+65+50+79+79=323 books removed. Now, however, we need to remove only one book because we will know that we have only two kinds of books left (either chemistry or physics) and any of them will give us a set of 80. Of course in reality it would not be that bad, but we have to take the worst situation to be sure. So, the answer is 324 (E).

 

Resume.  Today’s lesson covers a specific type of combinatorical problems that requires an unusual way of solving. You should know right and fast solution, like the solutions described in the lesson, in order to finish the GMAT on time.

 

zuevaMaterial prepared by Ksenia Zueva,
GMAT/GRE Quantitative consultant at MBA Strategy.

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