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A company hires 13 new employees and they are to be placed in New York (4 of them), London (5 of them), Tokyo (4 of them). How many ways can this placement be done ?

To go along with this... Bob works in the same company. Bob wants to stay in New York.

How many ways can this happen ?

Please help! Not sure how to attack this problem...

2 Answers 2

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It is a multinomial coefficient. You choose 4 out of 13 and place them in NY, then you choose 5 of the remaining 9 and place them in London, and the rest of 4 you place in Tokyo:

$(4,5,4)!=\frac{13!}{4!5!4!}={13\choose 4}{9\choose 5}{4\choose 4}$

If Bob stays in New York, you have only 12 employees to distribute: 3 in NY, 5 in London and 4 in Tokyo

$(3,5,4)!=\frac{12!}{3!5!4!}={12\choose 3}{9\choose 5}{4\choose 4}$

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    when I did the above math, I got 37065600 for the first question and 3193344 for the second question. Both were wrong. Am I still doing something wrong? – 2017-01-29
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    It is too much. I get 90090 at the first and 27720 at the second. – 2017-01-29
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    Okay I redid it on an actual calculator and thats what I got – 2017-01-29
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For New York we have 13 choices for 4 positions , for London 9 choices for 5 positions (as we substract 4 people we sent to New York) and the rest go to Tokyo. Thus the answer is ${{13}\choose{4}} * {{9}\choose{5}}$ . For the second question just substract one person from the 13 employees and bring the available positions for New York down to 3. Thus the answer is ${{12}\choose{3}}*{{9}\choose{5}}$

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    Also when I do it this way, I got 784604620800 for the first part and 241416806400 for the second part. Both were also wrong – 2017-01-29