Permutations Of Digits

Forming numbers with a size restriction is handled by constraining the leading digit first, then arranging the rest, a frequent JEE Advanced pattern. A four-digit number using the distinct digits 1, 2, 3, 4, 5 without repetition must exceed 3000, so its thousands digit must be at least 3, giving 3 choices: 3, 4, or 5. After fixing the leading digit, the remaining three places are filled by any 3 of the remaining 4 digits in P(4,3) = 4·3·2 = 24 ways. By the multiplication principle, the total is 3 × 24 = 72. Option 120 = 5·4·3·2 ignores the size restriction. Option 60 uses an incorrect leading-digit count. Option 96 overcounts the leading choices. Hence there are 72 such numbers. Plausibility check: the unrestricted count of four-digit numbers from these digits is P(5,4) = 120, and exactly three of the five possible leading digits qualify, so the restricted count is (3/5)·120 = 72, consistent with the leading-digit argument. Magnitude conditions on formed numbers are managed by first fixing the most significant digit to satisfy the size constraint, then freely arranging the remaining digits. This leading-digit-first strategy generalizes to divisibility and range problems, where the controlling position is identified and constrained before the multiplication principle is applied to the comparatively free remaining places.