Big Balls — Problem Completed
6 balls into 3 boxes, each at most 3 balls. Total unrestricted: (\binom{6+2}{2}=28). Subtract cases where some box ≥ 4. Let ( y_i = x_i - 4 \ge 0 ). For one box fixed: ( y_1 + x_2 + x_3 = 2 ) → (\binom{2+2}{2}=6). For 3 boxes: (3\times 6 = 18). Two boxes ≥4: impossible (8>6). Inclusion-exclusion: (28 - 18 = 10).
This is the “Big Balls Problem” — each box starts with ( m ) big balls, then distributes the rest. If each ( x_i \le M ), use inclusion-exclusion: [ \sum_{j=0}^k (-1)^j \binom{k}{j} \binom{n - j(M+1) + k - 1}{k - 1} ] where terms with negative top vanish. 4. Variants and Generalizations 4.1 Distinct Balls into Distinct Boxes If balls are distinguishable: ( k^n ) ways. 4.2 Indistinguishable Balls, Indistinguishable Boxes (Partitions) Number = ( p_k(n) ) = number of partitions of ( n ) into at most ( k ) parts (no simple closed form; uses generating functions). 4.3 “Big Balls” with Odd/Even Constraints E.g., each box gets an even number of balls. Generating function: [ (1 + x^2 + x^4 + \dots)^k = \frac{1}{(1-x^2)^k} ] Coefficient of ( x^n ) = ( \binom{n/2 + k - 1}{k - 1} ) if ( n ) even, else 0. 4.4 Balls of Two Sizes (Big and Small) Given ( b ) big balls (size ( B )) and ( s ) small balls (size 1), total volume ( n = bB + s ), distribute volume into ( k ) boxes. This becomes a bivariate generating function problem. 5. Worked Examples Example 1 (Standard big balls): 10 identical balls into 4 boxes, each at least 2 balls. Solution: ( n' = 10 - 4\times 2 = 2 ), ( k = 4 ) → (\binom{2+4-1}{3} = \binom{5}{3} = 10). big balls problem completed
often imposes lower bounds: each box must contain at least ( m ) balls (with ( m ) possibly large, hence “big balls”). 3. Core Mathematical Formulation 3.1 Stars and Bars (No Constraints) Number of non-negative integer solutions to ( x_1 + x_2 + \dots + x_k = n ): [ \binom{n + k - 1}{k - 1} ] 3.2 Lower Bound Constraints Let each ( x_i \ge m ). Set ( y_i = x_i - m \ge 0 ). Then: [ y_1 + \dots + y_k = n - km ] Number of solutions: [ \binom{n - km + k - 1}{k - 1}, \quad \text{if } n \ge km; \text{ else } 0. ] 6 balls into 3 boxes, each at most 3 balls
