Question

In how many ways 11 identical toys be placed in 3 distinct boxes such that no box is empty?

• A. 33
• B. 44
• C. 42
• D. 72
  (E) 45
• E. 45

Hint:

"No box is empty" is a very common phrase in these problems. It simply means $x \ge 1$. Use the simpler n-1 formula to save time!

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
"No box is empty" means each box must contain at least 1 toy. This is the condition for finding natural number solutions.
Step 2: Key Formula or Approach:
The number of ways to distribute $n$ identical items into $r$ distinct boxes such that each box gets at least one is: \[ ^{n-1}C_{r-1} \]
Step 3: Detailed Explanation:
1. Total toys ($n$) = 11.
2. Total boxes ($r$) = 3.
3. Apply the formula: \[ ^{11-1}C_{3-1} = \, ^{10}C_2 \] 4. Calculate the value: \[ ^{10}C_2 = \frac{10 \times 9}{2 \times 1} = 45. \]
Step 4: Final Answer:
The number of ways to place the toys is 45.