Question

In how many ways can 8 identical pens be distributed among Aal, Bal, and Cal so that Aal gets at least 1 pen, Bal gets at least 2 pens, and Cal gets at least 3 pens?

• A. 4
• B. 2
• C. 3
• D. 6
• E. 5

Hint:

For distribution of identical items with minimum constraints, reduce the variables to non-negative and use stars and bars.

Answer 1

The Correct Option is D

Step 1: Let Aal get $a$, Bal get $b$, Cal get $c$, with $a \ge 1$, $b \ge 2$, $c \ge 3$, and $a + b + c = 8$.
Step 2: Let $a' = a - 1 \ge 0$, $b' = b - 2 \ge 0$, $c' = c - 3 \ge 0$. Then $a' + b' + c' = 8 - (1+2+3) = 2$.
Step 3: Number of non-negative integer solutions to $a' + b' + c' = 2$ is $\binom{2+3-1}{3-1} = \binom{4}{2} = 6$.
Step 4: Final Answer: 6.