Using the digits 3, 4, 5, 6 and 7, and repeating no digit within a number, how many 4-digit numbers greater than 5000 can be formed?

A 4-digit number built from these digits clears 5000 exactly when its leading digit is 5, 6 or 7 — 3 choices. The remaining three places are then filled from the 4 unused digits in $4\times 3\times 2=24$ ways. Total = $3\times 24=72$. Trap $120=5\times 4\times 3\times 2$ counts every 4-digit number and ignores the size condition. Trap 48 admits only 6 and 7 as leading digits, but even the smallest number starting with 5 is 5346, which already beats 5000. Trap 24 fixes just one leading digit.