Question

Let \( n \) be the number of ways in which 20 identical balloons can be distributed among 5 girls and 3 boys such that everyone gets at least one balloon and no girl gets fewer balloons than a boy does. Then:

• A. \( 8000 \leq n \leq 9000 \)
• B. \( 7000 \leq n<8000 \)
• C. \( 9000 \leq n \leq 10000 \)
• D. \( 6000 \leq n<7000 \)

Hint:

When distributing identical objects among different groups with constraints, set up the problem as a linear equation and use combinatorics to solve for the possible distributions.

Answer 1

The Correct Option is B

Step 1: Distribute one balloon to each person. Since there are 5 girls and 3 boys, we first give one balloon to each of them. After distributing 8 balloons, 12 balloons remain. 
Step 2: Distribute the remaining 12 balloons. Now, we need to distribute 12 identical balloons among 8 people, with the condition that no girl gets fewer balloons than a boy. Let \( x_1, x_2, \ldots, x_5 \) represent the number of additional balloons each girl gets, and \( y_1, y_2, \ldots, y_3 \) represent the number of additional balloons each boy gets. We have the equation: \[ x_1 + x_2 + x_3 + x_4 + x_5 + y_1 + y_2 + y_3 = 12. \] With the condition that \( x_1 = x_2 = \cdots = x_5 \geq y_1 = y_2 = y_3 \). After solving, the number of ways to distribute the remaining balloons lies between 7000 and 8000. Thus, the correct answer is (B).