Distributing Identical Pens with Minimum Requirements

CAT 2017 Slot 2 · QA · Medium · Permutation and Combination

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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?

Answer

6

Explanation

First, give the minimum required pens to each person:

  • Amal gets 1 pen
  • Bimal gets 2 pens
  • Kamal gets 3 pens

Total pre-distributed pens = 1+2+3=61 + 2 + 3 = 6 pens.

Remaining pens to distribute = 8−6=28 - 6 = 2 identical pens among 3 people.

Number of ways using Stars and Bars formula (n+k−1k−1)\binom{n + k - 1}{k - 1}: (2+3−13−1)=(42)=6 ways\binom{2 + 3 - 1}{3 - 1} = \binom{4}{2} = 6\text{ ways}.

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