Question

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 1

Let Amal = A; Bimal = B; Kamal = K
Converting words into numbers we get,
A + B + K = 8,
Also, A ≥ 1; B ≥ 2; K ≥ 3
Since the pens are identical, we can give the least required pens to all.
Finally, we get the equation as:
A + B + K = 2
Now, this becomes the case of distributing n identical things among r persons when each person can get any number of things.
Hence, number of ways possible = $n+r-1_{C_{r-1}}=2+3-1_{C_{3-1}}=4_{C_2}=6$ ways.