Lovers Meet - Part 8

Anna and Ben live at opposite corners of a $m \times n$ block.
If Anna only moves right or up or down, and isn't allowed to retrace her steps, how many ways does she have to get to Ben?

#Combinatorics

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

m^(n-1)

Mobin Mithun - 5 years, 11 months ago

Why?

Calvin Lin Staff - 5 years, 11 months ago

(m+1)^n it is?

Shib Shankar Sikder - 5 years, 10 months ago

Start at the beginning. Anna can move right or down. If she moves down, she can only move down until she moves right. She can move to the second column in n+1 ways. The same case occurs for every column. Anna will have n+1 ways of moving to the next column each time. This keeps going until she is in the last column, in which case she has to move toward Ben. There are m+1 columns, each of which give n+1 ways of moving to the right, except for the last column. Therefore there m^(n+1) ways for Anna to reach Ben.

A Former Brilliant Member - 5 years, 10 months ago

Yes, that's the quick way to recognize it. Once you choose the horizontal paths (one in each column), then the vertical paths are fixed. This is why we can set up the bijection.

Chung Kevin - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$