Question

The number of ways in which 17 apples can be distributed among four guests such that each guest gets at least 3 apples is:

• A. \( 1140 \)
• B. \( 336 \)
• C. \( 36 \)
• D. \( 56 \)

Hint:

The "stars and bars" method is a powerful combinatorial tool for distributing indistinguishable objects into distinguishable bins, especially useful when there are no restrictions on how many objects each bin can contain.

Answer 1

The Correct Option is D

Step 1: First ensure each guest receives 3 apples, allocating \( 3 \times 4 = 12 \) apples and leaving \( 17 - 12 = 5 \) apples to be distributed. 
Step 2: Utilize the "stars and bars" theorem to distribute the remaining 5 apples among 4 guests without any restrictions.
- The formula for distributing \( n \) identical items among \( k \) groups is given by: \[ \binom{n+k-1}{k-1} \] - For our case, \( n = 5 \) apples and \( k = 4 \) guests, the formula becomes: \[ \binom{5+4-1}{4-1} = \binom{8}{3} \] Step 3: Calculate \( \binom{8}{3} \) which simplifies to: \[ \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]