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How-many-even-numbers-between-3000-and-7000-can-be-formed-using-the-digits-2-3-4-5-6-and-8-if-repetition-of-digits-is-not-allowed-




Question Number 883 by 112358 last updated on 12/Apr/15
How many even numbers   between 3000 and 7000 can be  formed using the digits   2,3,4,5,6 and 8 if repetition  of digits is not allowed?
$${How}\:{many}\:{even}\:{numbers}\: \\ $$$${between}\:\mathrm{3000}\:{and}\:\mathrm{7000}\:{can}\:{be} \\ $$$${formed}\:{using}\:{the}\:{digits}\: \\ $$$$\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\:{and}\:\mathrm{8}\:{if}\:{repetition} \\ $$$${of}\:{digits}\:{is}\:{not}\:{allowed}? \\ $$
Answered by prakash jain last updated on 12/Apr/15
First digits 3,5  =2        Last digit (2,4,6,8)=4        Middle two digits^4 P_2 =((4!)/(2!))=12         Total with first digits 3,5=2×4×12=96  First digit 4,6=2        Last digits (2,6,8) or (4,6,8)=3        Middle two digits^4 P_2 =((4!)/(2!))=12         Total with first digits 4,6=2×3×12=72  Total =72+96=168
$$\mathrm{First}\:\mathrm{digits}\:\mathrm{3},\mathrm{5}\:\:=\mathrm{2} \\ $$$$\:\:\:\:\:\:\mathrm{Last}\:\mathrm{digit}\:\left(\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{8}\right)=\mathrm{4} \\ $$$$\:\:\:\:\:\:\mathrm{Middle}\:\mathrm{two}\:\mathrm{digits}\:^{\mathrm{4}} \mathrm{P}_{\mathrm{2}} =\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{12} \\ $$$$\:\:\:\:\:\:\:\mathrm{Total}\:\mathrm{with}\:\mathrm{first}\:\mathrm{digits}\:\mathrm{3},\mathrm{5}=\mathrm{2}×\mathrm{4}×\mathrm{12}=\mathrm{96} \\ $$$$\mathrm{First}\:\mathrm{digit}\:\mathrm{4},\mathrm{6}=\mathrm{2} \\ $$$$\:\:\:\:\:\:\mathrm{Last}\:\mathrm{digits}\:\left(\mathrm{2},\mathrm{6},\mathrm{8}\right)\:\mathrm{or}\:\left(\mathrm{4},\mathrm{6},\mathrm{8}\right)=\mathrm{3} \\ $$$$\:\:\:\:\:\:\mathrm{Middle}\:\mathrm{two}\:\mathrm{digits}\:^{\mathrm{4}} \mathrm{P}_{\mathrm{2}} =\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{12} \\ $$$$\:\:\:\:\:\:\:\mathrm{Total}\:\mathrm{with}\:\mathrm{first}\:\mathrm{digits}\:\mathrm{4},\mathrm{6}=\mathrm{2}×\mathrm{3}×\mathrm{12}=\mathrm{72} \\ $$$$\mathrm{Total}\:=\mathrm{72}+\mathrm{96}=\mathrm{168} \\ $$
Commented by 112358 last updated on 12/Apr/15
Thanks!
$${Thanks}! \\ $$

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