Palmira Lochridge: Suppose that there are eight corporations competing for nine different contracts. If the contracts are awarded randomly, what is the probability that each corporation will get at least one contract?
Idell Mulliniks: The contracts may be awarded in 8^9 ways, and we may use inclusion-exclusion to remove the ways that 1, 2, 3, ... 7 corporations get 0 contracts.1 corporation gets 0 contracts in C(8,1)7^9 ways, 2 corporations get 0 contracts in C(8,2)6^9 ways etc.So the ways in which the 8 corporations get at least 1 contract each is:8^9 - C(8,1)7^9 + C(8,2)6^9 - C(8,3)5^9 + C(8,4)4^9 - C(8,5)3^9 + C(8,6)2^9 - C(8,7)1^9so that the probability is given by ÷8^9:prob =1 - C(8,1)(7/8)^9 + C(8,2)(6/8)^9 - C(8,3)(5/8)^9 + C(8,4)(4/8)^9 - C(8,5)(3/8)^9 + C(8,6)(2/8)^9 - C(8,7)(1/8)^9= 2835 / 262144= 0.0108 (4 d.p)(around 10%)...Show more
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