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Question Number 36844 by tawa tawa last updated on 06/Jun/18
find the sum of 4 digit even numbers formed from the digit 1, 2, 3, 4
$$\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{4}\:\mathrm{digit}\:\mathrm{even}\:\mathrm{numbers}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{the}\:\mathrm{digit}\:\:\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4} \\ $$
Answered by MJS last updated on 06/Jun/18
4×4×4×2=128 even numbers 64 abc2 64 abc4 32 ab1c 32 ab2c ... 32(1000+2000+3000+4000)=320000 32(100+200+300+400)=32000 32(10+20+30+40)=3200 64(2+4)=384 sum=355584
$$\mathrm{4}×\mathrm{4}×\mathrm{4}×\mathrm{2}=\mathrm{128}\:\mathrm{even}\:\mathrm{numbers} \\ $$$$\mathrm{64}\:\mathrm{abc2} \\ $$$$\mathrm{64}\:\mathrm{abc4} \\ $$$$ \\ $$$$\mathrm{32}\:\mathrm{ab1c} \\ $$$$\mathrm{32}\:\mathrm{ab2c} \\ $$$$… \\ $$$$ \\ $$$$\mathrm{32}\left(\mathrm{1000}+\mathrm{2000}+\mathrm{3000}+\mathrm{4000}\right)=\mathrm{320000} \\ $$$$\mathrm{32}\left(\mathrm{100}+\mathrm{200}+\mathrm{300}+\mathrm{400}\right)=\mathrm{32000} \\ $$$$\mathrm{32}\left(\mathrm{10}+\mathrm{20}+\mathrm{30}+\mathrm{40}\right)=\mathrm{3200} \\ $$$$\mathrm{64}\left(\mathrm{2}+\mathrm{4}\right)=\mathrm{384} \\ $$$$ \\ $$$$\mathrm{sum}=\mathrm{355584} \\ $$
Commented by Rasheed.Sindhi last updated on 06/Jun/18
Nice Sir!
$$\mathcal{N}{ice}\:{Sir}! \\ $$
Commented by tawa tawa last updated on 06/Jun/18
please what is 32 ab1c etc. from your solution. i can see that numbers that end with 2 is 64, numbers that end with 4 is 64. i don′t understand the 32 ab1c etc thanks. please explain
$$\mathrm{please}\:\mathrm{what}\:\mathrm{is}\:\:\mathrm{32}\:\mathrm{ab1c}\:\:\:\mathrm{etc}.\:\:\mathrm{from}\:\mathrm{your}\:\mathrm{solution}.\:\mathrm{i}\:\mathrm{can}\:\mathrm{see}\:\mathrm{that}\:\mathrm{numbers}\:\mathrm{that} \\ $$$$\mathrm{end}\:\mathrm{with}\:\mathrm{2}\:\mathrm{is}\:\mathrm{64},\:\mathrm{numbers}\:\mathrm{that}\:\mathrm{end}\:\mathrm{with}\:\mathrm{4}\:\mathrm{is}\:\mathrm{64}.\:\:\:\mathrm{i}\:\mathrm{don}'\mathrm{t}\:\mathrm{understand}\:\mathrm{the}\:\mathrm{32}\:\mathrm{ab1c}\:\mathrm{etc} \\ $$$$\mathrm{thanks}.\:\mathrm{please}\:\mathrm{explain} \\ $$
Commented by tawa tawa last updated on 06/Jun/18
And what if repetition of digit is not allowed
$$\mathrm{And}\:\mathrm{what}\:\mathrm{if}\:\mathrm{repetition}\:\mathrm{of}\:\mathrm{digit}\:\mathrm{is}\:\mathrm{not}\:\mathrm{allowed} \\ $$
Commented by tawa tawa last updated on 06/Jun/18
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by MJS last updated on 06/Jun/18
abc stand for any combination of 1, 2, 3, 4 we have 128 numbers, each 32 start with 1000, 2000, 3000, 4000, each 32 contain 100, 200, 300,400, each 32 contain 10, 20, 30, 40 if no repetition is allowed: 1234, 1324, 2134, 2314, 3124, 3214 1342, 1432, 3142, 3412, 4132, 4312 sum=31116
$${abc}\:\mathrm{stand}\:\mathrm{for}\:\mathrm{any}\:\mathrm{combination}\:\mathrm{of}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4} \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{128}\:\mathrm{numbers},\:\mathrm{each}\:\mathrm{32}\:\mathrm{start}\:\mathrm{with} \\ $$$$\mathrm{1000},\:\mathrm{2000},\:\mathrm{3000},\:\mathrm{4000},\:\mathrm{each}\:\mathrm{32}\:\mathrm{contain} \\ $$$$\mathrm{100},\:\mathrm{200},\:\mathrm{300},\mathrm{400},\:\mathrm{each}\:\mathrm{32}\:\mathrm{contain}\:\mathrm{10},\:\mathrm{20},\:\mathrm{30},\:\mathrm{40} \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{no}\:\mathrm{repetition}\:\mathrm{is}\:\mathrm{allowed}: \\ $$$$\mathrm{1234},\:\mathrm{1324},\:\mathrm{2134},\:\mathrm{2314},\:\mathrm{3124},\:\mathrm{3214} \\ $$$$\mathrm{1342},\:\mathrm{1432},\:\mathrm{3142},\:\mathrm{3412},\:\mathrm{4132},\:\mathrm{4312} \\ $$$$\mathrm{sum}=\mathrm{31116} \\ $$
Commented by tawa tawa last updated on 06/Jun/18
can we get that 31116 by calculation ?. i used your method above. am not getting 31116. i appreciate your effort. God bless you sir
$$\mathrm{can}\:\mathrm{we}\:\mathrm{get}\:\mathrm{that}\:\mathrm{31116}\:\mathrm{by}\:\mathrm{calculation}\:?.\:\:\mathrm{i}\:\mathrm{used}\:\mathrm{your}\:\mathrm{method}\:\mathrm{above}.\:\mathrm{am}\:\mathrm{not} \\ $$$$\mathrm{getting}\:\mathrm{31116}. \\ $$$$ \\ $$$$\mathrm{i}\:\mathrm{appreciate}\:\mathrm{your}\:\mathrm{effort}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by MJS last updated on 06/Jun/18
4×1000+2×2000+4×3000+2×4000=28000 4×100×2×200+4×300+2×400=2800 4×10+2×20+4×30+2×40=280 6×2+6×4=36 28000+2800+280+36=31116
$$\mathrm{4}×\mathrm{1000}+\mathrm{2}×\mathrm{2000}+\mathrm{4}×\mathrm{3000}+\mathrm{2}×\mathrm{4000}=\mathrm{28000} \\ $$$$\mathrm{4}×\mathrm{100}×\mathrm{2}×\mathrm{200}+\mathrm{4}×\mathrm{300}+\mathrm{2}×\mathrm{400}=\mathrm{2800} \\ $$$$\mathrm{4}×\mathrm{10}+\mathrm{2}×\mathrm{20}+\mathrm{4}×\mathrm{30}+\mathrm{2}×\mathrm{40}=\mathrm{280} \\ $$$$\mathrm{6}×\mathrm{2}+\mathrm{6}×\mathrm{4}=\mathrm{36} \\ $$$$\mathrm{28000}+\mathrm{2800}+\mathrm{280}+\mathrm{36}=\mathrm{31116} \\ $$
Commented by tawa tawa last updated on 06/Jun/18
wow great but i dont really understand. i want to understand so that i can know it because of exam
$$\mathrm{wow}\:\mathrm{great}\:\mathrm{but}\:\mathrm{i}\:\mathrm{dont}\:\mathrm{really}\:\mathrm{understand}.\:\mathrm{i}\:\mathrm{want}\:\mathrm{to}\:\mathrm{understand}\:\mathrm{so}\:\mathrm{that}\:\mathrm{i}\:\mathrm{can}\:\mathrm{know}\: \\ $$$$\mathrm{it}\:\mathrm{because}\:\mathrm{of}\:\mathrm{exam}\: \\ $$

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