Question

4 boys picked up 30 mangoes. In how many ways can they divide them, if all mangoes be identical (a) \({ }^{33} \mathrm{C}_{4}\) (b) \({ }^{33} \mathrm{C}_{2}\) (c) 5456 (d) 6554 .

Short answer

The number of ways 4 boys can divide 30 identical mangoes among themselves is 5456, which corresponds to option (c).

Step-by-step solution

Recognize the stars and bars technique

The problem can be solved using the stars and bars technique, which is used to partition items among groups (in this case, boys).

Calculate the total number of slots available

There are 30 mangoes to be distributed, and as there are 4 boys, the mangoes will be separated by 3 "bars" (representing the divisions among the boys). Thus, there are a total of 33 slots to be filled by either stars (mangoes) or bars (boys).

Select the slots for the bars

As there are 3 bars to be placed in 33 slots, we can use the combination formula to find the number of ways to place these bars in the slots. The combination formula is: \[C(n,r) = \frac{n!}{r!(n-r)!}\] Where n is the total number of slots available, and r is the number of bars to be placed.

Calculate the combination

Using the combination formula from step 3, plug in n = 33 and r = 3: \[C(33,3) = \frac{33!}{3!(33-3)!}\]

Simplify the formula

By simplifying the combination formula, we get the final answer: \[ \begin{aligned} C(33,3) &=\frac{33!}{3!(30)!} \\ &=\frac{33 \times 32 \times 31}{3 \times 2 \times 1} \\ &=5456 \end{aligned} \] The answer is 5456, which corresponds to option (c).

Key concepts

Stars and Bars Technique

The stars and bars technique is a clever method for solving combinatorial problems involving the distribution of identical items into distinct groups. In the context of the exercise, we have 30 identical mangoes (the stars) that need to be divided among 4 boys (the partitions represented by bars).

This technique works by considering the mangoes as stars and the separations between different groups of mangoes as bars. For instance, if you have 2 stars followed by a bar, followed by 5 more stars, you are essentially saying that the first boy gets 2 mangoes, and the second boy gets 5. Each configuration of stars and bars represents a different way to distribute the items.

To determine the overall number of distributions, imagine laying out 33 slots in a row. These slots can either be filled with mangoes (stars) or divides (bars), and the total number of ways to arrange the bars among the stars gives the solution to the problem.

Combination Formula

The combination formula is used to determine the number of ways to select a subset from a larger set, where the order of selection does not matter. In this problem, it's applied to find out how many ways we can place the bars among the slots.

The formula for combination is given by:

• \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Here,

• \(n\) is the total number of items to choose from, which is 33 in our case, accounting for both mangoes and the spaces for bars.
• \(r\) is the number of items to choose, which corresponds to the number of bars, that is 3.

Utilizing the combination formula is essential as it provides the exact count of configurations without missing any possibilities. It reveals that there are 5456 different ways to arrange the stars and bars, aligning with option (c).

Partitioning Objects

Partitioning objects, especially when they are identical like mangoes in this problem, can be complex yet logical when approached correctly. Using partition techniques helps to distribute identical objects among distinct groups fairly easily without having to manually list each possible outcome.

This concept leverages the idea of breaking down a larger task (like dividing mangoes) into smaller, countable steps (allocating partitions).

By establishing partitions with bars, we essentially transform the problem into finding how different ways these partitions could be arranged. This allows us to utilize formulas and combinatorial principles to quickly arrive at the solution, making the task of counting partitions less daunting and more systematic.