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Question

Difficulty: EasyPermutations

In a science competition, 33 distinct prizes (first, second, and third place) are to be awarded to 33 different students chosen from a group of 55 finalists. In how many different ways can these 33 prizes be awarded?

Answer: 60 ways

Answer

60 ways
Because the prizes are distinct, the order in which the finalists are selected matters. Calculating the number of arrangements of 33 students from 55 candidates is given by 5P3=5×4×3=60^{5}P_{3} = 5 \times 4 \times 3 = 60.

Step-by-Step Solution

1
Determine if order matters
Since the prizes are distinct (1st, 2nd, and 3rd place), the order of assignment matters, making this a permutation problem.
Assigning distinct positions to individuals requires calculating permutations rather than combinations.
2
Apply the permutation formula nPr=n!(n−r)!^{n}P_{r} = \frac{n!}{(n-r)!}
5P3=5!(5−3)!=5!2!^{5}P_{3} = \frac{5!}{(5-3)!} = \frac{5!}{2!}
There are 55 total candidates (n=5n = 5) and 33 positions to fill (r=3r = 3).
3
Calculate the numeric value
5P3=5×4×3=60^{5}P_{3} = 5 \times 4 \times 3 = 60
Canceling 2!2! from the numerator and denominator leaves 5×4×3=605 \times 4 \times 3 = 60.

Key Concept

Linear permutation of r items selected from n distinct items
Estimated Time:45s
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Topics

Permutations

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