SOLUTION: 9 different pies are to be divided between 3 people so that each person gets an odd number of pies. Find the number of ways this can be done

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Question 1149517: 9 different pies are to be divided between 3 people so that each person gets an odd number of pies. Find the number of ways this can be done
Answer by greenestamps(13198) About Me  (Show Source):
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If each gets an odd number of pies, then the numbers of pies are either 1, 3, and 5 (in some order); or 3, 3, and 3.

Give each of the three people his allotment of pies, one person at a time.

(1) First person gets 5 of the 9; second gets 3 of the remaining 4; third gets 1 of the remaining 1. C%289%2C5%29%2AC%284%2C3%29%2AC%281%2C1%29+=+126%2A4%2A1 = 504 ways
(2) First person gets 5 of the 9; second gets 1 of the remaining 4; third gets 3 of the remaining 3. C%289%2C5%29%2AC%284%2C1%29%2AC%283%2C3%29+=+126%2A4%2A1 = 504 ways
(3) First person gets 3 of the 9; second gets 5 of the remaining 6; third gets 1 of the remaining 1. C%289%2C3%29%2AC%286%2C5%29%2AC%281%2C1%29+=+84%2A6%2A1 = 504 ways
(4) First person gets 3 of the 9; second gets 3 of the remaining 6; third gets 3 of the remaining 3. C%289%2C3%29%2AC%286%2C3%29%2AC%283%2C3%29+=+84%2A20%2A1= 1680 ways
(5) First person gets 3 of the 9; second gets 1 of the remaining 6; third gets 5 of the remaining 5. C%289%2C3%29%2AC%286%2C1%29%2AC%285%2C5%29+=+84%2A6%2A1 = 504 ways
(6) First person gets 1 of the 9; second gets 5 of the remaining 8; third gets 3 of the remaining 3. C%289%2C1%29%2AC%288%2C5%29%2AC%283%2C3%29+=+9%2A56%2A1 = 504 ways
(7) First person gets 1 of the 9; second gets 3 of the remaining 8; third gets 5 of the remaining 5. C%289%2C1%29%2AC%288%2C3%29%2AC%285%2C5%29+=+9%2A56%2A1 = 504 ways

TOTAL: 1680+6(504) = 4704 ways