Three-digit numbers from S = {2, 3, 4, 5, 7, 9} less than 400: How many distinct-digit three-digit numbers < 400 can be formed?

Introduction / Context:We form three-digit numbers from a given set with distinct digits and a numeric bound (< 400). The hundreds place determines the bound handling; then we count permutations for the remaining places.

Given Data / Assumptions:S = {2, 3, 4, 5, 7, 9}; digits cannot repeat; number < 400.

Concept / Approach:Constrain the hundreds digit first. Only 2 or 3 keep the number below 400. For each choice of hundreds, count permutations of tens and units from the remaining digits.

Step-by-Step Solution:

Verification / Alternative check:Enumerating quickly for hundreds=2 and 3 confirms symmetry and count.

Why Other Options Are Wrong:20 counts only one hundreds choice; 80 or 120 overcount beyond the < 400 constraint.

Common Pitfalls:Accidentally including 4,5,7,9 as hundreds, which violates the bound.

Final Answer:40