How many five digit numbers that are divisible by 4 can be formed by using the digits 0 to 7 if no digit occurs more than once in any number

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For numbers to be divisible by 4, the last two digits should be divisible by 4.The numbers formed from digits 0−7 should have 04,12,16,20,24,32,36,40,52,56,60,64,72,76 as last two digits to be divisible by 4.The total possible cases are =141) When the last two digits don't have a zero that is 10 casesThen the left most digit will have 5 options since 0 cannot be at the leftmost digit(otherwise it will become a 4 digit number)The next digit will also have 5 options(including 0)The next place will have only 4 options since digits cannot be repeated.Hence the number of possible ways are =10×5×5×4=10002) When the last two digits  have a zero that is 4 casesThen the left most digit will have 6 options since 0 already used.The next digit will also have 5 optionsThe next place will have only 4 options since digits cannot be repeated.Hence the number of possible ways are =4×6×5×4=480Therefore the answer is 1000+480=1480

Question Text	How many five digit numbers that are divisible by $4$ can be formed by using the digits $0$ to $7$ if no digit occurs more than once in any number
Answer Type	Text solution:1
Upvotes	150

How many five digit numbers that are divisible by $4$ can be formed by using the digits $0$ to $7$ if no digit occurs more than once in any number

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