Combinations With Identical Items

Distributing identical objects into distinct groups with a minimum-per-group condition uses the stars and bars method after reserving the minimums, a standard JEE Advanced technique. First give each of the 3 children one chocolate to satisfy the at-least-one condition, using 3 chocolates and leaving 7 to distribute freely. The number of ways to distribute 7 identical chocolates among 3 children with no restriction is C(7 + 3 - 1, 3 - 1) = C(9, 2) = 36. The stars and bars formula counts non-negative integer solutions of x_1 + x_2 + x_3 = 7. Option 66 = C(12,2) forgets to reserve the minimums. Option 120 overcounts. Option 45 = C(10,2) uses the wrong total. Hence there are 36 ways. Plausibility check: the distribution corresponds to placing 2 dividers among 7 identical items, and C(9,2) = 36 counts these divider placements, consistent with each child ending up with at least one chocolate. The multinomial correction, dividing the total factorial by the factorial of each repetition count, removes the indistinguishable rearrangements created by identical letters. This same coefficient appears in the multinomial theorem expansion, so the counting of arrangements of a multiset and the algebra of multinomial coefficients are two views of one combinatorial identity that JEE Advanced repeatedly exploits.