psm12se wrote:In city A, the streets are aligned in a grid, where the east-west roads are called 1st Rd, 2nd Rd, 3rd Rd, etc, increasing in number as one moves northward. The north-south roads are called 1st Ave, 2nd Ave, 3rd Ave, etc, increasing in number as one moves eastward. If Bill needs to walk from the corner of 2nd Rd and 3rd Ave to the corner of 6th Rd and 8th Ave in the shortest possible time, how many different routes could he take?
A. 2772
B. 126
C. 117
D. 84
E. 9
OA: B
A quick refresher on permutations with IDENTICAL ELEMENTS:
The number of ways to arrange the letters AABBB = 5!/2!3!.
The total number of ways to arrange 5 elements (5!) is divided by 2! to account for the 2 identical A's and by 3! to account for the 3 identical B's.
To travel from 2nd Rd. and 3rd Ave. to 6th Rd. and 8th Ave., 5 blocks must be traveled eastward and 4 blocks must be traveled northward.
Let the 5 blocks traveled eastward = EEEEE.
Let the 4 blocks traveled northward = NNNN.
Any arrangement of the letter EEEEENNNN represents a possible route, since any arrangement of these letters represents exactly 5 blocks traveled eastward and 4 blocks traveled northward.
To illustrate:
The dark line in the figure above shows one possible route: NNEEEEENN.
Thus, the number of possible routes is equal to the number of ways to arrange the letters EEEEENNNN:
9!/5!4! = 126.
The correct answer is
B.
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