Distribution Into Distinct Boxes

Distributing distinct objects into distinct boxes with no restriction gives each object an independent choice of box, an application of the multiplication principle in JEE Advanced. Each of the 5 distinct letters can independently go into any of the 3 mailboxes, providing 3 choices per letter. By the multiplication principle, the total number of distributions is 3^5 = 243. The exponent is the number of objects and the base is the number of boxes, the canonical form (boxes)^(objects). Option 125 = 5^3 reverses base and exponent. Option 15 = 5 × 3 wrongly adds choices. Option 10 = C(5,2) confuses this with a selection. Hence there are 243 ways. Plausibility check: this distinct-into-distinct count equals the number of functions from a 5-set to a 3-set, which is 3^5 = 243, consistent with each letter being assigned exactly one mailbox. A combination equation conceals a quadratic in the unknown count, and spotting the product of consecutive integers usually resolves it faster than formal factoring. These equations recur whenever a geometric or selection count is specified and the underlying population size is sought, so fluency in reversing C(n,2) and C(n,3) values into n is a worthwhile exam reflex.