A general in command of 105 troops is issued orders to send all his

Question

A general in command of 105 troops is issued orders to send all his men, excluding himself, into combat. His directives are as follows: 1) To limit casualties, he cannot send everyone in one wave; 2) For each wave after the first, he must send exactly half as many troops as there were in the previous wave.

How many unique ways are there for the general to deploy his troops within the directives (e.g., the invalid combination 50-25 would count as one)?

A) 0
B) 2
C) 3
D) 4
E) 9

Self-made question.

(Confession: I just wanted to write a fun question that encouraged critical reasoning, not necessarily one that would appear on the GMAT™.)

INCORRECT = MISSION FAILURE  (total loss of life)    TIME    OUTCOME   
    CORRECT —Pyrrhic victory   5:00+  76 percent loss of life (80 troops)  
    CORRECT —Strategic victory   3:30-4:59  50 percent loss of life (53 troops)  
    CORRECT —Decisive victory   2:01-3:29  25 percent loss of life (26 troops)  
    CORRECT —Total victory   Up to 2:00  10 percent loss of life (11 troops)  
   Note:  Some loss of life is inevitable in battle.

AndrewN · Answer

Hello, everyone. This has to be my least popular self-made question, but I guess you cannot win them all. Perhaps the timing of my post was off, or I could have done a better job with the presentation, making the question more GMAT™-like. Whatever the case, I think the Number Properties concepts—e.g., divisibility—are quite applicable.

You can start by factoring 105. Its prime factorization is 3 * 5 * 7, so these values might provide a good starting point for exploration. We also know that each wave after the first will be half the size of the previous wave, so we might want to think of the first wave not as, say,  x , but as 2 x , or 4 x , to avoid dealing with unnecessary fractions. Consider 3:

2x + x = 105

3x = 105

x = 35

This works. If the first wave had 70 troops and the second had 35, then all 105 troops would have been sent into combat, per the orders. We can keep doubling the coefficient at the start of the equation to check for other successful combinations.

4x + 2x + x = 105

7x = 105

x = 15

This also works. If the first wave consisted of 60 troops, the second 30, and the last one 15, then the general has followed orders. Keep going.

8x + 4x + 2x + x = 105

15x = 105

x = 7

Another valid combination. 56 troops were sent into combat, followed by 28, then 14, then 7. What happens if we double the coefficient once more?

16x + 8x + 4x + 2x + x = 105

31x = 105

No integer value will work, so we know this five-step combination is invalid. There is only one more coefficient to test within reason.

32x + 16x + 8x + 4x + 2x + x = 105

63x = 105

Not only will this equation also not yield an integer value, but we can see that there is no need to start with 64 x  in a 7-tiered system: 64 + 32 + 16 is already greater than 105.

Thus, the only ways in which the general can send his troops into battle within the given constraints are the following:

70-35   60-30-15   56-28-14-7

Since we have three valid combinations,   the answer must be (C)  . I had a lot of fun with this one. As always, good luck with your studies.

- Andrew

Sajjadmahessar · Answer

I am struggling witht these sort of questions, any idea what is the name of the topic which includes these types of questions?

AndrewN · Answer

Hello,  Sajjadmahessar . First off, please note that this was designed to be a question for amusement, so you are not likely to see one similar in presentation on the GMAT™. That said, I would recommend studying theory on the following if you find the question challenging:

Word problems   Divisibility/Multiples/Fractions   Number Properties

Despite the  how many ways  of the question stem, this is  not  a probability/combinatorics question. Rather, I designed it with logic and basic arithmetic in mind.

If you need to study more theory, I would recommend checking the pertinent sections of the Ultimate GMAT Quantitative Megathread and ALL YOU NEED FOR QUANT ! ! ! both of which can be found by scrolling down  the main page of the Quantitative thread .

I hope you had fun with the question. Good luck with your studies.

- Andrew

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A general in command of 105 troops is issued orders to send all his

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INCORRECT = MISSION FAILURE (total loss of life)	TIME	OUTCOME
CORRECT—Pyrrhic victory	5:00+	76 percent loss of life (80 troops)
CORRECT—Strategic victory	3:30-4:59	50 percent loss of life (53 troops)
CORRECT—Decisive victory	2:01-3:29	25 percent loss of life (26 troops)
CORRECT—Total victory	Up to 2:00	10 percent loss of life (11 troops)
Note: Some loss of life is inevitable in battle.

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A general in command of 105 troops is issued orders to send all his

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My Rewards