Question

In how many ways 5 identical envelopes be placed among 3 distinct boxes such that any number of envelopes can be placed in any box.

• A. 21
• B. 35
• C. 6
• D. 42
  (E) None of the above
• E. None of the above

Hint:

Always identify if the items are identical or distinct. If they are identical (like these envelopes), use the Stars and Bars formula. If they were distinct, the formula would be $r^n$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a problem of distributing identical items into distinct groups, which is solved using the "Stars and Bars" method. Since any number of envelopes can be placed in any box, we are looking for non-negative integer solutions (whole numbers).

Step 2: Key Formula or Approach:
The number of ways to distribute $n$ identical items into $r$ distinct boxes is: \[ ^{n+r-1}C_{r-1} \]

Step 3: Detailed Explanation:
1. Number of identical envelopes ($n$) = 5.
2. Number of distinct boxes ($r$) = 3.
3. Apply the formula: \[ ^{5+3-1}C_{3-1} = \, ^7C_2 \] 4. Calculate the value: \[ ^7C_2 = \frac{7 \times 6}{2 \times 1} = 21. \]

Step 4: Final Answer:
The number of ways to place the envelopes is 21.