kullayappayenugula wrote:Pat will walk from Intersection X to Intersection Y
along a route that is confined to the square grid of
four streets and three avenues shown in the map
above. How many routes from X to Y can Pat take
that have the minimum possible length?
Diagram attached....
(A) 6
(B) 8
(C) 10
(D) 14
(E) 16
How to solve this using P&C?[/img]
Let E = traveling one block eastward and N = traveling one block northward.
To travel from X to Y by a route of minimum length, Pat must travel exactly 2 blocks eastward and exactly 3 blocks northward:
EENNN.
Any arrangement of the letters EENNN will yield a viable route.
The number of ways to arrange 5 DISTINCT elements = 5!.
But the elements above are not distinct: there are 2 identical E's and 3 identical N's.
When an arrangement includes IDENTICAL elements, we must DIVIDE by the number of ways to arrange the identical elements.
The reason is that the arrangement doesn't change when the identical elements swap positions, REDUCING the number of unique arrangements:
To illustrate:
The number of ways to arrange the letters in the word SPEED = 5!/2! = 60.
We divide by 2! to account for the 2 E's.
The number of ways to arrange the letters in the word RADAR = 5!/(2!*2!) = 30.
We divide by 2! to account for the 2 A's and by another 2! to account for the 2 R's.
The number of ways to arrange the letters in the word MISSISSIPPI = 11!/(4!*4!*2!).
We divide by 4! to account for the 4 S's, by another 4! to account for the 4 I's, and by 2! to account for the 2 P's.
In the problem at hand:
The number of ways to arrange EENNN = 5!/(2!*3!) = 10.
The correct answer is
C.
Check here for a similar -- but trickier -- problem:
https://www.beatthegmat.com/different-routes-t93698.htmlPrivate tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at
[email protected].
Student Review #1
Student Review #2
Student Review #3
