Question-155257

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

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

the grid has m×n steps. find the number of routes with k turns.

$${the}\:{grid}\:{has}\:\boldsymbol{{m}}×\boldsymbol{{n}}\:{steps}. \\ $$$${find}\:{the}\:{number}\:{of}\:{routes}\:{with}\:\boldsymbol{{k}} \\ $$$${turns}. \\ $$

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

it is to find the number of routes with exactly k turns. the example shows a route with 7 turns. the grid is generally of size m×n.

$${it}\:{is}\:{to}\:{find}\:{the}\:{number}\:{of}\:{routes} \\ $$$${with}\:{exactly}\:{k}\:{turns}.\:{the}\:{example} \\ $$$${shows}\:{a}\:{route}\:{with}\:\mathrm{7}\:{turns}.\:{the}\:{grid} \\ $$$${is}\:{generally}\:{of}\:{size}\:{m}×{n}. \\ $$

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)

$$\:\mathrm{C}_{\:\mathrm{8}} ^{\:\mathrm{18}} =\:\mathrm{C}_{\mathrm{10}} ^{\mathrm{18}} =\:\mathrm{43758} \\ $$$$\mathrm{required}\:\mathrm{number}\:\mathrm{of}\:\mathrm{routes} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{taking} \\ $$$$\mathrm{m}\:\mathrm{right}\:\mathrm{turns}\:\left(\mathrm{or}\:\mathrm{n}\:\mathrm{down}\:\mathrm{turns}\right) \\ $$

Commented by mathdanisur last updated on 28/Sep/21

bemath = benjo qoarto gultom) this solution does not apply to you.. this solution #lekraj beedassy is the solution..

$$\left.\boldsymbol{\mathrm{b}}\mathrm{emath}\:=\:\boldsymbol{\mathrm{b}}\mathrm{enjo}\:\boldsymbol{\mathrm{q}}\mathrm{oarto}\:\boldsymbol{\mathrm{g}}\mathrm{ultom}\right) \\ $$$$\mathrm{this}\:\mathrm{solution}\:\mathrm{does}\:\mathrm{not}\:\mathrm{apply}\:\mathrm{to}\:\mathrm{you}.. \\ $$$$\mathrm{this}\:\mathrm{solution}\:#\boldsymbol{\mathrm{l}}\mathrm{ekraj}\:\boldsymbol{\mathrm{b}}\mathrm{eedassy}\:\mathrm{is}\:\mathrm{the}\:\mathrm{solution}.. \\ $$

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

$${please}\:{explain}\:{what}\:{you}\:{mean}! \\ $$

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)

$$\mathrm{If}\:{n}+{m}−{k}>\mathrm{0}: \\ $$$$\mathrm{there}\:\mathrm{are}\:\mathrm{2}.\left({k}+\mathrm{1}\right)^{{n}+{m}−\left({k}+\mathrm{1}\right)} \:\mathrm{routes} \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{not}\:\mathrm{there}\:\mathrm{arn}'\mathrm{t}\:\mathrm{any}. \\ $$$$ \\ $$$$\left(\mathrm{proof}\:\mathrm{down}\:\mathrm{below},\:\mathrm{sorry}\:\mathrm{for}\:\mathrm{not}\:\mathrm{typing}\:\mathrm{it}\right. \\ $$$$\mathrm{but}\:\mathrm{I}\:\mathrm{feel}\:\mathrm{like}\:\mathrm{the}\:\mathrm{drawings}\:\mathrm{help}\:\mathrm{the}\: \\ $$$$\left.\mathrm{understanding}\right) \\ $$

Commented by TheHoneyCat last updated on 29/Sep/21

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

for k=1 there are two routes. but you get 2×(1+1)^(n+m−2) =2^(n+m−1) !

$${for}\:{k}=\mathrm{1}\:{there}\:{are}\:{two}\:{routes}.\:{but}\:{you} \\ $$$${get}\:\mathrm{2}×\left(\mathrm{1}+\mathrm{1}\right)^{{n}+{m}−\mathrm{2}} =\mathrm{2}^{{n}+{m}−\mathrm{1}} ! \\ $$

Commented by TheHoneyCat last updated on 29/Sep/21

Damn it! You are right... I have forgotten that the constraint on length and turns is not an equivalence... I am counting both: →↓↓↓ & →→↓↓ My method can still be modified without a lot of work to get the correct solution I must make a distinction betwee “→” and “↓” But I am quite too tired to do it now. (I live in Europe) This weekend maybe... Sorry for not double checking

$$\mathrm{Damn}\:\mathrm{it}!\:\mathrm{You}\:\mathrm{are}\:\mathrm{right}… \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{have}\:\mathrm{forgotten}\:\mathrm{that}\:\mathrm{the}\:\mathrm{constraint}\:\mathrm{on} \\ $$$$\mathrm{length}\:\mathrm{and}\:\mathrm{turns}\:\mathrm{is}\:\mathrm{not}\:\mathrm{an}\:\mathrm{equivalence}… \\ $$$$\mathrm{I}\:\mathrm{am}\:\mathrm{counting}\:\mathrm{both}: \\ $$$$\rightarrow\downarrow\downarrow\downarrow\:\&\:\rightarrow\rightarrow\downarrow\downarrow \\ $$$$ \\ $$$$\mathrm{My}\:\mathrm{method}\:\mathrm{can}\:\mathrm{still}\:\mathrm{be}\:\mathrm{modified}\:\mathrm{without} \\ $$$$\mathrm{a}\:\mathrm{lot}\:\mathrm{of}\:\mathrm{work}\:\mathrm{to}\:\mathrm{get}\:\mathrm{the}\:\mathrm{correct}\:\mathrm{solution} \\ $$$$\mathrm{I}\:\mathrm{must}\:\mathrm{make}\:\mathrm{a}\:\mathrm{distinction}\:\mathrm{betwee}\:“\rightarrow'' \\ $$$$\mathrm{and}\:“\downarrow''\: \\ $$$$\mathrm{But}\:\mathrm{I}\:\mathrm{am}\:\mathrm{quite}\:\mathrm{too}\:\mathrm{tired}\:\mathrm{to}\:\mathrm{do}\:\mathrm{it}\:\mathrm{now}. \\ $$$$\left(\mathrm{I}\:\mathrm{live}\:\mathrm{in}\:\mathrm{Europe}\right) \\ $$$$\mathrm{This}\:\mathrm{weekend}\:\mathrm{maybe}… \\ $$$$ \\ $$$$\mathrm{Sorry}\:\mathrm{for}\:\mathrm{not}\:\mathrm{double}\:\mathrm{checking} \\ $$

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