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

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

$$thegridhas\mathit{m}\times \mathit{n}steps.$$$$findthenumberofrouteswith\mathit{k}$$$$turns.$$

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

$$itistofindthenumberofroutes$$$$withexactlykturns.theexample$$$$showsaroutewith7turns.thegrid$$$$isgenerallyofsizem\times n.$$

Commented by bemath last updated on 28/Sep/21

$${\mathrm{C}}_8^{18}={\mathrm{C}}_{10}^{18}=43758$$$$\mathrm{required}\mathrm{number}\mathrm{of}\mathrm{routes}$$$$\mathrm{is}\mathrm{a}\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{of}\mathrm{taking}$$$$\mathrm{m}\mathrm{right}\mathrm{turns}\left(\mathrm{or}\mathrm{n}\mathrm{down}\mathrm{turns}\right)$$

Commented by mathdanisur last updated on 28/Sep/21

$$\mathbf{b}\mathrm{emath}=\mathbf{b}\mathrm{enjo}\mathbf{q}\mathrm{oarto}\mathbf{g}\mathrm{ultom})$$$$\mathrm{this}\mathrm{solution}\mathrm{does}\mathrm{not}\mathrm{apply}\mathrm{to}\mathrm{you}..$$$$\text{You can't use 'macro parameter character \#' in math mode}$$

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

$$pleaseexplainwhatyoumean!$$

Answered by TheHoneyCat last updated on 29/Sep/21

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

Commented by TheHoneyCat last updated on 29/Sep/21

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

$$fork=1therearetworoutes.butyou$$$$get2\times {(1+1)}^{n+m-2}=2^{n+m-1}!$$

Commented by TheHoneyCat last updated on 29/Sep/21

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

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