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Chapter 5: Problem 439

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

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

Step by step solution

01

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).
02

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).
03

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.
04

Calculate the combination

Using the combination formula from step 3, plug in n = 33 and r = 3: \[C(33,3) = \frac{33!}{3!(33-3)!}\]
05

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).

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