To solve the problem of finding the number of ways to place 3 identical balls into 4 distinct boxes (with boxes allowed to be empty), we use the stars and bars combinatorial method. Here's the step-by-step reasoning:

Key Insight: Stars and Bars Theorem

When distributing (n) identical items into (k) distinct boxes (allowing empty boxes), the number of ways is given by the combination formula:
[ \binom{n + k - 1}{k - 1} ]

This formula arises because:

• We represent the (n) identical balls as stars ((*)).
• We need (k-1) bars ((| )) to separate the stars into (k) groups (one per box).
• The total number of positions to arrange the stars and bars is (n + k -1), and we choose (k-1) positions for the bars (the rest are stars).

Application to the Problem

• (n = 3) (identical balls), (k =4) (distinct boxes).
• Compute the combination:
  [ \binom{3 +4 -1}{4-1} = \binom{6}{3} ]

Calculation of (\binom{6}{3}):
[ \binom{6}{3} = \frac{6!}{3! \cdot (6-3)!} = \frac{6 \times5 \times4}{3 \times2 \times1} = 20 ]

Answer: 20. The number of ways is 20, derived from the stars and bars formula for distributing identical objects into distinct boxes with no restrictions.

(\boxed{20})

作者声明:本文包含人工智能生成内容。