What is the number of the shortest routes from X to Y through Z?

Question

What is the number of the shortest routes from X to Y through Z? 12.jpg A. 8 B. 12 C. 16 D. 18 E. 20

BrentGMATPrepNow · Accepted Answer

Nice question!

Let's first get from  point X to point Z . 
NOTE: we're trying to get to point Z via the SHORTEST route. 
So, one possible route is: Up, Up, Right, Right (UURR)
Another possible route is: Right, Right, Up, Up (RRUU)
Another possible route is: Right, Up, Up, Right (RUUR)
etc

As you can see, the shortest route from point X to point Z will consist of 2 Ups (U's) and 2 Rights (R')
So, the question really becomes, "In how many different ways can we arrange 2 U's and 2 R's?"

-------ASIDE---------------------------------
When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:

If there are  n  objects where  A  of them are alike, another  B  of them are alike, another  C  of them are alike, and so on, then the total number of possible arrangements =  n !/ A !)( B !)( C !)....]

So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows: 
There are  11  letters in total 
There are  4  identical I's
There are  4  identical S's
There are  2  identical P's
So, the total number of possible arrangements =  11 !/ 4 !)( 4 !)( 2 !)]
--------------------------------------------------------------

To get from point X to point Z, we must arrange 2 U's and 2 R's
There are  4  letters in total 
There are  2  identical U's
There are  2  identical R's
So, the total number of possible arrangements =  4 !/ 2 !)( 2 !) =  6

So, there are  6  different routes  from point X to point Z .

Once we arrive at point Z, we can see that there are only   2   possible ways to get to point Y: 
1) Up then right
2) Right then up

So, for each of the  6  different routes to get to point Z, there are 2 ways to get to point Y
So, the TOTAL number of ways to get from point X to point Y (via point Z) =  6  x   2   = 12

Answer: B

MathRevolution · Answer

==> The number of cases of x heading to z is 4!/2!2!=6. The number of cases of Z heading to y is 2, which makes 6*2=12.

Hence, the answer is B.
Answer: B

HouseStark · Answer

No matter what, you have to take 2 horizontal steps and 2 vertical steps to reach point "Z" from "X".

let's Horizontal step=H
vertical step= V

"HHVV"    the no. of ways to arrange them=  \frac{4!}{2!*2!} = 6   ways to reach Z

the no. of ways to reach  Y  from  Z  only  2 ways

now, multiply the results=  6*2= 12   ways total

Answer B

abuc0112 · Answer

If I see a question like this again, how do I tell it is a combination question?

MathRevolution · Answer

=> Best Approach: Regard the different parts of the shortest route as repeated letters in a word
=> Start at X, End at Y, Stop at Z

Two parts: Shortest route (X to Z) & Shortest route (Z to Y)

Modify the question: X to Z
 Stretches of vertical roads/ steps = ‘a’ 
 Stretches of horizontal roads/ steps = ‘b’
=> List all stretches of roads: b,b,a,a

=> Total number of roads: 4 (2 b^′ s & 2 a^′ s)

=> "∴" Formula: 4!/(2! 2! ) = 6

=> Modify the question: Z to Y
 Stretches of vertical roads/steps = ‘p’ 
 Stretches of horizontal roads/steps = ‘q’

=> List all stretches of roads: p,q

=> Total number of roads: 2 (1 p^′ s & 1 q^′ s)

=> " ∴" Formula: 2! = 2

=> Shortest routes: 6 routes (X~Z) and 2 routes (Z~Y)

=> " ∴" Total possible shortest routes: 6 * 2 = 12

Thanks

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the number of the shortest routes from X to Y through Z?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the number of the shortest routes from X to Y through Z?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards