Question-155257

Question Number 155257 by mr W last updated on 27/Sep/21

Commented by mr W last updated on 28/Sep/21

t h e g r i d h a s m \times n s t e p s .

f i n d t h e n u m b e r o f r o u t e s w i t h k

t u r n s .

Commented by mr W last updated on 28/Sep/21

i t i s t o f i n d t h e n u m b e r o f r o u t e s

w i t h e x a c t l y k t u r n s . t h e e x a m p l e

s h o w s a r o u t e w i t h 7 t u r n s . t h e g r i d

i s g e n e r a l l y o f s i z e m \times n .

Commented by bemath last updated on 28/Sep/21

C_{8}^{18} = C_{10}^{18} = 43758

required number of routes

is a number of ways of taking

m right turns (or n down turns)

Commented by mathdanisur last updated on 28/Sep/21

b emath = b enjo q oarto g ultom)

this solution does not apply to you . .

You can't use 'macro parameter character #' in math mode

Commented by mr W last updated on 28/Sep/21

p l e a s e e x p l a i n w h a t y o u m e a n!

Answered by TheHoneyCat last updated on 29/Sep/21

If n + m - k > 0 :

there are 2 . {(k + 1)}^{n + m - (k + 1)} routes

if not there {arn}^{'} t any .

(proof down below, sorry for not typing it

but I feel like the drawings help the

understanding)

Commented by TheHoneyCat last updated on 29/Sep/21

Commented by mr W last updated on 29/Sep/21

f o r k = 1 t h e r e a r e t w o r o u t e s . b u t y o u

g e t 2 \times {(1 + 1)}^{n + m - 2} = 2^{n + m - 1}!

Commented by TheHoneyCat last updated on 29/Sep/21

Damn it! You are right \dots

I have forgotten that the constraint on

length and turns is not an equivalence \dots

I am counting both :

\to↓↓↓ & \to\to↓↓

My method can still be modified without

a lot of work to get the correct solution

I must make a distinction betwee “ \to^{″}

and “ ↓^{″}

But I am quite too tired to do it now .

(I live in Europe)

This weekend maybe \dots

Sorry for not double checking