In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens?

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6 — Easy First hand out the guaranteed minimums, then freely distribute whatever pens remain using the stars-and-bars formula. 1 Give out the minimums first. Amal, Bimal and Kamal need at least $1$, $2$ and $3$ pens. $$\begin{aligned} &	ext{pens left}=8-(1+2+3)=2 \end{aligned}$$ 2 Distribute the remaining pens freely. Now $2$ identical pens go to $3$ people with no restriction. The number of non-negative integer solutions of $x_1+x_2+x_3=2$ is given by stars and bars. $$\begin{aligned} &\binom{2+3-

CAT 2017 Slot 2 — QA Question 29

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