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quiz1_key - ⋅ 5-7(5 ⋅ 2 =252-21=231 Alternative way is...

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COT3100 Summer’2001 Quiz#1(25 pts) Solutions Leave answers in factorial form! 1) (5pts). A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark. How many socks must he take out to be sure that he has at least two socks of the same color? 3 2) (10pts). Three women and seven men are on the faculty of CS department. The individuals are distinguishable. How many ways are there to select a committee of 5 members if at least one woman must be on the committee? There are C (7, 5) committees that do not contain women. Subtracting this number from the total number C (10,5) committees gives the answer. 10!/(5!
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Unformatted text preview: ⋅ 5!)-7!/(5! ⋅ 2!) =252-21=231 Alternative way is to count directly committees with 1, 2 or three women. The total number will be: C (7,4) ⋅ C (3,1)+ C (7,3) ⋅ C (3,2)+ C (7,2) ⋅ C (3,3)= 3 ⋅ 7!/(4! ⋅ 3!)+3 ⋅ 7!/(3! ⋅ 4!)+7!/(2! ⋅ 5!)= 3 ⋅ 35+3 ⋅ 35+21=231. 3) (10pts). In how many ways a card hand consisting of 3 spades and 2 hearts can be selected from a standard 52-card deck? 3 spades can be selected in C (13,3) different ways. 2 hearts can be selected in C (13,2) different ways. By the product rule the total number of hands is C (13,3) ⋅ C (13,2)=(13 ⋅ 12 ⋅ 11) ⋅ (13 ⋅ 12)/(3 ⋅ 2 ⋅ 2)=286 ⋅ 78...
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