A teacher wants to give 5 prizes to a class 27 students. I low many ways can he do this if all the prizes are the same and students can receive more that one prize?
How many ways can he do this if all the prizes are different and students can receive more that one prize?
How many ways can he do this if all the prizes are the same and students can not receive more that one prize?
How many ways can he do this Wall the prizes are different and students can not receive more that one prize?
To find the number of ways the teacher can give the prizes, we can use the concept of combinations and permutations.
1. If all the prizes are the same and students can receive more than one prize:
In this case, we are essentially distributing the identical prizes among the students. This situation can be modeled as distributing indistinguishable items into distinguishable containers with repetition allowed. The formula to calculate this is (n + r - 1) choose (r), where n is the number of students (27) and r is the number of prizes (5). Therefore, the number of ways to distribute the prizes is (27 + 5 - 1) choose (5) = 31 choose 5 = 8,435.
2. If all the prizes are different and students can receive more than one prize:
Now, we have different prizes to distribute. For each prize, the teacher has 27 choices of who to give it to, and since students can receive more than one prize, each choice is independent. Therefore, for each prize, there are 27 possibilities. Since there are 5 prizes in total, the total number of ways to distribute the prizes is 27^5 = 14,348,907.
3. If all the prizes are the same and students cannot receive more than one prize:
In this case, each student can only receive one prize, so we are essentially selecting 5 students out of 27 to receive the prizes. This situation can be modeled as selecting a subset (combination) of 5 students out of a total of 27. The formula to calculate this is 27 choose 5 = 84,285.
4. If all the prizes are different and students cannot receive more than one prize:
Similar to the previous case, we need to select 5 students out of 27 to receive the prizes. Using the combination formula, this can be calculated as 27 choose 5 = 84,285.
In summary:
1. Identical prizes, students can receive more than one prize: 8,435 ways
2. Different prizes, students can receive more than one prize: 14,348,907 ways
3. Identical prizes, students cannot receive more than one prize: 84,285 ways
4. Different prizes, students cannot receive more than one prize: 84,285 ways