Scholarship Awarding Problem Explanation
This problem involves finding the number of ways to distribute 3 distinct scholarships among 4 applicants, where each applicant can receive one or more scholarships.
Mathematical Approach
Consider each scholarship individually:
- For the first scholarship, there are 4 possible applicants it can be awarded to.
- For the second scholarship, since an applicant can receive multiple scholarships, there are again 4 possible applicants.
- Similarly, for the third scholarship, there are 4 possible applicants.
Since the awarding of each scholarship is an independent event, we multiply the number of choices for each scholarship to find the total number of ways.
Calculation
The total number of ways is calculated as:
Number of ways = (Choices for Scholarship 1) $\times$ (Choices for Scholarship 2) $\times$ (Choices for Scholarship 3)
Number of ways = $4 \times 4 \times 4 = 4^3$
Number of ways = $64$
Conclusion
There are 64 different ways to award the 3 scholarships to the 4 applicants under the given condition.
Important Questions from Permutations and Combinations
- How many 4-digit positive integers divisible by 3 can be formed using only the digits {1, 3, 4, 6, 7}, such that no digit appears more than once in a number?
Three distinct sets of indistinguishable twins are to be seated at a circular table that has 8 identical chairs. Unique seating arrangements are defined by the relative positions of the people.
How many unique seating arrangements are possible such that each person is sitting next to their twin?
- Mixed species flocks of birds include social and solitary species. There are 5 social species and 10 solitary species in a forest. Flocks always have a total of 5 species, of which 2 are social and 3 are solitary. The number of types of flocks with unique species composition is ________.(Answer in integer)
- How many five-digit numbers can be formed using the integers 3, 4, 5 and 6 with exactly one digit appearing twice?
Two wizards try to create a spell using all the four elements, water, air, fire, and earth. For this, they decide to mix all these elements in all possible orders. They also decide to work independently. After trying all possible combination of elements, they conclude that the spell does not work.
How many attempts does each wizard make before coming to this conclusion, independently?
