In how many ways can we distribute 10 identical pencil into 4 students so that…

To solve this problem, we use the stars and bars method for distributing identical items with constraints.

Step 1: Since each student must get at least 1 pencil, give 1 pencil to each of the 4 students. This uses up 4 pencils.

Step 2: We are left with 10 - 4 = 6 pencils to distribute freely among the 4 students.

Step 3: The number of ways to distribute 6 identical pencils among 4 students (where students can get zero or more) is given by the formula:

C(n + r - 1, r - 1) = C(6 + 4 - 1, 4 - 1) = C(9, 3)

Step 4: Calculate C(9, 3) = 84.

Thus, the number of ways is 84.