How many numbers greater than $56000$ can be formed by using the digit $4,5,6,7,8$ with no digit repeated.

To form numbers greater than $56000$ using the digits $4,5,6,7,8$ without repetition can be done

(1)Taking first digit as 5 OR

(2) first digit greater than 5

(1) If the first digit is 5 can be done in 1 way The second digit without repetition can be done in 3 ways ( it can not be 4) Rest of the 3 digits can be done in 3! = 6 ways Therefore Taking first digit as 5 numbers greater than 56000 can be formed =$1x3x6=18$ OR

(2) First digit is greater than $5$ The first digit can be formed in 3 ways. Rest of the 4 digits can be done in $4!=24$ Therefore Taking first digit > 5 numbers greater than $56000$ can be formed = 3 x 24 = 72 Hence Numbers greater than $56000,$ using the digits $4,5,6,7,8$ without repeating$=18+72=90$