Question

In how many ways can we distribute 10 identical looking pencils to 4 students so that each student gets at least one pencil?

• A. 5040
• B. 210
• C. 84
• D. None of these
• E.

Hint:

When distributing identical objects with restrictions, use the stars and bars method after accounting for the given constraints (e.g., each student getting at least one object).

Answer 1

The Correct Option is C

This problem involves distributing identical objects (pencils) among distinct groups (students) while ensuring that each student receives at least one pencil. To apply the stars and bars method, we first allocate one pencil to each student. This reduces the total pencils to be distributed from 10 to 6. Now, we must distribute these 6 remaining pencils among 4 students without any restrictions. The number of ways to do this is given by the stars and bars formula: \[ \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3} = 84 \]