How many ways can 5 prizes be distributed among four students when every student can take one or more prizes 1024?

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Answer

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Hint: In the given question, we are provided with five prizes to be distributed to four students such that every student can take any number of prizes. So, we first find the number of options with each prize to be allocated to any person. Then, we use the multiplication rule of counting the total number of ways in which prizes can be distributed.

Complete step-by-step solution:

So, we have $5$ prizes to be distributed among four students when every student can take one or more prizes. So, we calculate the number of options for each of the five prizes for the allocation. Since, each of the four students can be assigned any number of prizes. So, we have the number of options with the first prize to be allocated to the person as $5$. Similarly, all the five prizes will be allocated to one of the four students. Hence, the number of options with each of the five prizes to be distributed is equal to the number of students, that is $5$.Now, we know the number of ways of allocation of individual prizes. So, we will use the multiplication rule of counting. According to the multiplication rule of counting, if we have m ways to do a certain thing and n ways to do another thing, then there are a total of $m \times n$ ways to do both the things together. Since, we have to distribute all the prizes together. So, the number of ways to do so $ = 5 \times 5\times 5 \times 5$Now, expressing the product in form of powers, we get, $ = {5^4}$Now, doing the calculations and computing the power, we get, $ = 625$Hence, we get the total number of ways of distributing five prizes to four students as $625$.

So, the option (B) is the correct answer.

Note: In such a problem, one must know the multiplication rule of counting as consequent events are occurring. One must take care while doing the calculations and should recheck them so as to be sure of the final answer. We must have a good grip and accuracy over arithmetic.

In how many ways can 5 prizes be distributed among four students when every student can take one or more prizes

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