sequence-of-string-said-to-be-orderly-if-element-index-i-different-to-i-1-for-example-aba-has-orderly-value-2-abab-has-orderly-value-3-abaabb-has-orderly-value-3-if-there-are-7-a-and-13-b-exa

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Question Number 195666 by uchihayahia last updated on 07/Aug/23

$$$$$$sequenceofstringsaidtobeorderly$$$$ifelementindexidifferenttoi+1$$$$forexample$$$$abahasorderlyvalue2$$$$ababhasorderlyvalue3$$$$abaabbhasorderlyvalue3$$$$ifthereare7aand13b$$$$example$$$$aaaaaaabbbbbbbbbbbbbhasorderlyvalue1$$$$whatisthemeanofitsorderlyvalue$$$$forallpossiblesequences?$$$$$$

Commented by mr W last updated on 09/Aug/23

$$igot9.1inatoughway,seebelow.$$$$doyoualsohaveasolution?$$

Commented by uchihayahia last updated on 09/Aug/23

$$i{don}^{\prime }t,itriedsolvingitusingpython$$$$toomuchtimeneeded$$

Answered by mr W last updated on 09/Aug/23

$$7“a^{″}and13“b^{″}$$$$n=orderlyvalue$$$$n_{min}=1$$$$n_{max}=14$$$$infollowing$$$$Ameansaboxcontainingoneor$$$$moreletters“a^{″}$$$$Bmeansaboxcontainingoneor$$$$moreletters“b^{″}$$$$$$$$n=1:$$$$A\mid B\Rightarrow 1way$$$$B\mid A\Rightarrow 1way$$$$---2$$$$n=2:$$$$A\mid B\mid A\Rightarrow 6\times 1=6ways$$$$B\mid A\mid B\Rightarrow 1\times 12=12ways$$$$---18$$$$n=3:$$$$A\mid B\mid A\mid B\Rightarrow 6\times 12=72ways$$$$B\mid A\mid B\mid A\Rightarrow 6\times 12=72ways$$$$---144$$$$n=4:$$$$A\mid B\mid A\mid B\mid A\Rightarrow 15\times 12=180ways$$$$B\mid A\mid B\mid A\mid B\Rightarrow 6\times 66=396ways$$$$---576$$$$n=5:$$$$A\mid B\mid A\mid B\mid A\mid B\Rightarrow 15\times 66=990ways$$$$B\mid A\mid B\mid A\mid B\mid A\Rightarrow 15\times 66=990ways$$$$---1980$$$$n=6:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 20\times 66=1320ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 15\times 220=3300ways$$$$---4620$$$$n=7:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 20\times 220=4400ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 20\times 220=4400ways$$$$---8800$$$$n=8:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 15\times 220=3300ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 20\times 495=9900ways$$$$---13200$$$$n=9:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 15\times 495=7425ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 15\times 495=7425ways$$$$---14850$$$$n=10:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 6\times 495=2970ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 15\times 792=11880ways$$$$---14850$$$$n=11:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 6\times 792=4752ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 6\times 792=4752ways$$$$---9504$$$$n=12:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 1\times 792=792ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 6\times 924=5544ways$$$$---6336$$$$n=13:$$$$A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 1\times 924=924ways$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\Rightarrow 1\times 924=924ways$$$$---1848$$$$n=14:$$$$B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\mid A\mid B\Rightarrow 1\times 792=792ways$$$$---792$$$$$$$$mean=\frac{2\times 1+18\times 2+144\times 3+576\times 4+1980\times 5+4620\times 6+8800\times 7+13200\times 8+14850\times 9+14850\times 10+9504\times 11+6336\times 12+1848\times 13+792\times 14}{2+18+144+576+1980+4620+8800+13200+14850+14850+9504+6336+1848+792}$$$$=\frac{705432}{77520}=9.1✓$$$$$$$$check:$$$$numberoftotalpossibilities$$$$=\frac{20!}{7!\times 13!}=77520✓$$

Commented by uchihayahia last updated on 09/Aug/23

$$thanksyou,i^{\prime }lldomybesttounderstandyour$$$$answer.stillstudyingbasiccombinatrics$$

Commented by mr W last updated on 09/Aug/23

$$i^{\prime }llgivesomeexplanationformy$$$$solution.$$$$example:theorderlyvalueisn=4.$$$$thatmeansthereare4$$$$placeswherea“a^{″}anda“b^{″}are$$$$nexttoeachother.suchaplaceis$$$$markedas“{\mid }^{″}.wehavetwocases:$$$$case1:A\mid B\mid A\mid B\mid A$$$$todistribute7“a^{″}in3boxesthereare$$$$C_{7-1}^{3-1}=15ways$$$$todistribute13“b^{″}in2boxesthereare$$$$C_{13-1}^{2-1}=12ways$$$$\Rightarrow totally15\times 12=180ways$$$$case2:B\mid A\mid B\mid A\mid B$$$$todistribute7“a^{″}in2boxesthereare$$$$C_{7-1}^{2-1}=6ways$$$$todistribute13“b^{″}in3boxesthereare$$$$C_{13-1}^{3-1}=66ways$$$$\Rightarrow totally6\times 66=396ways$$$$thereforethereare180+396=576$$$$possibilitiesfortheorderlyvalue4.$$

Commented by uchihayahia last updated on 11/Aug/23

$$thanks,stilllotofworkiguess.iasked$$$$myfriendandtoldme9.1isthecorrect$$$$answer$$

Commented by mr W last updated on 11/Aug/23

$$anywaymyanswer9.1iscorrect.$$$$canyourfriendtellushowhesolved?$$

Commented by uchihayahia last updated on 12/Aug/23

$$he{didn}^{\prime }ttellme,hesaidtheanswer$$$$islongandlaborous.buttheideaisthesame$$

Commented by mr W last updated on 12/Aug/23

$$ialsothinkthereisnomoresimple$$$$waythanthatofmine.$$

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