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If a company gives out awards to the top 5 employees based on performance, and t

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Question: If a company gives out awards to the top 5 employees based on performance, and there are 20 employees, how many different combinations of award winners are possible?

Options:

  1. 15504
  2. 18648
  3. 20020
  4. 21000

Correct Answer: 20020

Solution: 

The number of combinations is calculated using the formula for combinations: C(20, 5) = 20! / (5!(20-5)!) = 15504.

If a company gives out awards to the top 5 employees based on performance, and t

Practice Questions

Q1
If a company gives out awards to the top 5 employees based on performance, and there are 20 employees, how many different combinations of award winners are possible?
  1. 15504
  2. 18648
  3. 20020
  4. 21000

Questions & Step-by-Step Solutions

If a company gives out awards to the top 5 employees based on performance, and there are 20 employees, how many different combinations of award winners are possible?
  • Step 1: Understand that we need to find out how many ways we can choose 5 employees from a group of 20.
  • Step 2: Recognize that this is a combinations problem because the order of the employees does not matter.
  • Step 3: Use the combinations formula, which is C(n, r) = n! / (r!(n - r)!), where n is the total number of employees and r is the number of employees to choose.
  • Step 4: In this case, n = 20 (total employees) and r = 5 (employees to choose).
  • Step 5: Plug the numbers into the formula: C(20, 5) = 20! / (5!(20 - 5)!)
  • Step 6: Simplify the formula: C(20, 5) = 20! / (5! * 15!)
  • Step 7: Calculate 20! (20 factorial) which is 20 × 19 × 18 × 17 × 16 × 15! (we can cancel 15! in the numerator and denominator).
  • Step 8: Now we have: C(20, 5) = (20 × 19 × 18 × 17 × 16) / (5 × 4 × 3 × 2 × 1).
  • Step 9: Calculate the numerator: 20 × 19 × 18 × 17 × 16 = 1860480.
  • Step 10: Calculate the denominator: 5 × 4 × 3 × 2 × 1 = 120.
  • Step 11: Divide the numerator by the denominator: 1860480 / 120 = 15504.
  • Step 12: Conclude that there are 15504 different combinations of award winners.
  • Combinations – The concept of combinations involves selecting items from a larger set where the order of selection does not matter.
  • Factorials – Factorials are used in the calculation of combinations, representing the product of all positive integers up to a given number.
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