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

Answer 1

The Correct Option is C

Distribution of Identical Objects - Stars and Bars Method 

This problem involves distributing identical objects (pencils) to distinct groups (students) with the condition that each student receives at least one pencil.

Step 1: Adjusting for Minimum Requirement

Each student must receive at least one pencil. Therefore, we first give each student one pencil, reducing the number of pencils to distribute from 10 to 6.

Step 2: Applying Stars and Bars

Now, we need to distribute 6 pencils among 4 students without restrictions. We use the stars and bars formula:

$$\binom{n + k - 1}{k - 1}$$

where:

• n = 6 (remaining pencils)
• k = 4 (students)

Applying the formula:

$$\binom{6+4-1}{4-1} = \binom{9}{3}$$

Step 3: Calculating the Combination

$$\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!}$$

$$= \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$$

Thus, the total number of ways to distribute the pencils is 84.