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?

• A. 3
• B. 5
• C. 4
• D. 6

Answer 1

The Correct Option is D

To find the number of ways to distribute 8 identical pens among Amal, Bimal, and Kamal according to the conditions given, let's follow these steps:

1. Let the number of pens Amal, Bimal, and Kamal receive be represented by \( x_1, x_2, \) and \( x_3 \) respectively. We have the conditions: \( x_1 \geq 1 \), \( x_2 \geq 2 \), and \( x_3 \geq 3 \).
2.  Convert these inequalities to equations by defining new variables:
   • \( x_1' = x_1 - 1 \) (so \( x_1' \geq 0 \))
   • \( x_2' = x_2 - 2 \) (so \( x_2' \geq 0 \))
   • \( x_3' = x_3 - 3 \) (so \( x_3' \geq 0 \))
3. Substituting \( x_1, x_2, x_3 \) in terms of \( x_1', x_2', x_3' \) into the equation \( x_1 + x_2 + x_3 = 8 \), we have:
   \[(x_1' + 1) + (x_2' + 2) + (x_3' + 3) = 8\] Simplifying, we get:
   \[x_1' + x_2' + x_3' = 2\]
4. Now, find the number of non-negative integer solutions for the equation \( x_1' + x_2' + x_3' = 2 \). This is a classic stars and bars problem, where the number of solutions is given by \(\binom{n+k-1}{k-1}\) where \( n \) is the total to be divided and \( k \) is the number of variables:
   \[\binom{2+3-1}{3-1} = \binom{4}{2} = 6\]
5. Thus, there are 6 ways to distribute the pens under the given conditions.