SOLUTION: A firm has an opening for 3 computer programmers and there are 20 applicants. In how many different ways can the job be filled?

Question 48481: A firm has an opening for 3 computer programmers and there are 20 applicants. In how many different ways can the job be filled?
Answer by longjonsilver(2297) (Show Source): You can put this solution on YOUR website!
this is slightly more tricky.

Lets assume this is just like the other questions you asked. I shall answer it that way and then explain the problem and then correct my answer.

So... 20 people apply for a job. How many could get the first place? Answer is any of the 20.
And the second place? Answer is 19
And the third place? Answer is 18

So we have 20*19*18.

However, this is way too large because of the nature of what is being asked. Lets assume that applicants A, B and C are the ones who actually get the 3 jobs.

From the 20*19*18 answer - this includes the permutations ABC, ACB, BAC, BCA, CAB, CBA.

However all these 6 variations are the same: A,B and C got the job in all 6 cases. So our 20*19*18 answer is too big by 6 times, since the combination ABC is an answer just once. In other words, the order does not matter here.

So, correctly we have (20*19*18)/3! where the 3! gives us 6 and the 3 comes from the fact that we have 3 "jobs" to allocate.

This is where Permutations and Combinations are difficult... spotting the difference and for me it is tricky too, but i have had 20 years practice. Best way is to take it slowly and think about the physical nature of what is being asked, like i have written here.

jon

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