Question:medium

The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

Updated On: Jan 15, 2026
Show Solution

Correct Answer: 84

Solution and Explanation

It is required to distribute 20 identical balloons among 4 children. Each child must receive an even number of balloons, and at least one balloon. The number of balloons for the ith child is represented as \(2x_i\), where \(x_i\) is a non-negative integer. The equation governing this distribution is \(2x_1 + 2x_2 + 2x_3 + 2x_4 = 20\). Dividing by 2 yields \(x_1 + x_2 + x_3 + x_4 = 10\). The constraint that each child receives at least 1 balloon implies \(x_i \ge 1\). By substituting \(y_i = x_i - 1\), where \(y_i \ge 0\), the equation transforms to \((y_1+1) + (y_2+1) + (y_3+1) + (y_4+1) = 10\), which simplifies to \(y_1 + y_2 + y_3 + y_4 = 6\). The number of non-negative integer solutions to this equation is determined using the stars and bars method. With \(n=6\) (the sum) and \(k=4\) (the number of variables), the number of solutions is given by \(\binom{n+k-1}{k-1} = \binom{6+4-1}{4-1} = \binom{9}{3} = 84\). Therefore, there are 84 ways to distribute the balloons.
Was this answer helpful?
0

Top Questions on Permutation and Combination

Questions Asked in CAT exam

The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.
View Solution