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Question Number 39846 by vajpaithegrate@gmail.com last updated on 12/Jul/18

$$\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{four}\mathrm{digit}\mathrm{even}\mathrm{numbers}\mathrm{that}\mathrm{can}\mathrm{be}\mathrm{formed}\mathrm{with}\mathrm{digits}0354\mathrm{without}\mathrm{repitation}$$

Commented by Rasheed.Sindhi last updated on 13/Jul/18

$$43536$$

Commented by vajpaithegrate@gmail.com last updated on 13/Jul/18

$$\mathrm{how}\mathrm{sir}\mathrm{pls}\mathrm{explain}$$

Answered by Rasheed.Sindhi last updated on 13/Jul/18

$$\because \mathrm{The}\mathrm{numbers}\mathrm{are}\mathrm{even}$$$$\therefore \mathrm{The}\mathrm{unit}\mathrm{place}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{with}0\mathrm{or}4$$$$\mathrm{Case}:---0$$$$\mathrm{When}\mathrm{unit}\mathrm{is}0$$$$\mathrm{tens}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}3\mathrm{ways}(3,4\mathrm{or}5)$$$$\mathrm{after}\mathrm{that}\mathrm{hundreds}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}2\mathrm{ways}$$$$\mathrm{and}\mathrm{thousands}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}1\mathrm{way}\mathrm{only}.$$$$\therefore \mathrm{Total}\mathrm{numbers}\mathrm{with}\mathrm{unit}0:$$$$1\times 3\times 2\times 1=6$$$$\mathrm{Case}:---4$$$$\mathrm{When}\mathrm{unit}\mathrm{is}4$$$$\mathrm{thousands}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}2\mathrm{ways}\left(3\mathrm{or}5\right)$$$$(\mathrm{It}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{be}0\mathrm{as}\mathrm{in}\mathrm{that}\mathrm{case}\mathrm{the}\mathrm{numbers}$$$$\mathrm{will}\mathrm{be}\mathrm{of}3\mathrm{digits})$$$$\mathrm{hundres}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}2\mathrm{ways}\mathrm{and}$$$$\mathrm{tens}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}1\mathrm{way}.$$$$\therefore \mathrm{Total}\mathrm{numbers}\mathrm{with}\mathrm{unit}4:$$$$1\times 2\times 2\times 1=4$$$$\mathrm{So}\mathrm{sum}\mathrm{of}\mathrm{units}\mathrm{of}\mathrm{all}\mathrm{the}10\mathrm{numbers}\mathrm{is}$$$$0\times 6+4\times 4=16(6\mathrm{units}\mathrm{\&}1\mathrm{carry}\mathrm{to}\mathrm{tens}.$$$$\mathrm{Case}:--0-$$$$\mathrm{When}\mathrm{tens}\mathrm{is}0\to \mathrm{unit}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}1\mathrm{way}\left(\mathrm{with}4\right)$$$$\to \mathrm{hundreds}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}2\mathrm{ways}\left(3\mathrm{or}5\right)$$$$\to \mathrm{thousands}\mathrm{can}\mathrm{be}\mathrm{filled}\mathrm{in}1\mathrm{way}.$$$$\therefore \mathrm{Total}\mathrm{numbers}\mathrm{with}\mathrm{tens}0$$$$1\times 1\times 2\times 1=2$$$$\mathrm{Case}:--4-$$$$1\times 1\times 2\times 1=2$$$$\mathrm{Case}:--3-$$$$2\times 1\times$$$$\mathrm{Continue}$$

Answered by Rasheed.Sindhi last updated on 13/Jul/18

$$5430+4530+5340+3540+4350$$$$+3450+5304+3504+5034+3054$$$$=43536$$$$\mathrm{Note}:\mathrm{The}\mathrm{numbers}\mathrm{have}\mathrm{been}\mathrm{achieved}$$$$\mathrm{by}\mathrm{tree}-\mathrm{diagram},\mathrm{which}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{be}\mathrm{drawn}$$$$\mathrm{by}\mathrm{forum}-\mathrm{keyboard}.$$

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