Question

Number of ways of distributing 16 identical oranges among 4 persons such that each one gets at least one orange is:

• A. 435
• B. 455
• C. 470
• D. 489

Hint:

For distribution of identical objects with minimum constraints, first satisfy the minimum condition and then apply the stars and bars method.

Answer 1

The Correct Option is B

Step 1: Apply the condition of minimum one orange.
Let each person receive at least one orange. Distribute 1 orange to each of the 4 persons. Remaining oranges \( = 16 - 4 = 12 \).
Step 2: Convert to a stars and bars problem.
Now distribute 12 identical oranges among 4 persons with no restriction. Number of solutions of \[ x_1 + x_2 + x_3 + x_4 = 12 \] is given by the formula \[ \binom{12 + 4 - 1}{4 - 1} \]
Step 3: Calculate the number of ways.
\[ \binom{15}{3} = 455 \]
Step 4: Final Answer.
Hence, the required number of ways is \[ \boxed{455} \]