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midterm1solutions

# midterm1solutions - MAT344H1F Midterm Exam Duration 1 Hour...

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MAT344H1F Mid–term Exam, October 28, 2008 Duration: 1 Hour 45 Minutes No Aids Allowed This exam has 8 questions, for a total of 100 points. Answer the questions in the spaces provided. If you run out of room for an answer, continue on the back of the page. Please read the questions carefully. Be sure to justify your answer. An answer by itself will not get any credit. Last Name: First Name: Student ID: Question Points Score 1 10 2 10 3 15 4 10 5 15 6 15 7 10 8 15 Total: 100

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1. How many ways are there to form a committee of 5 people from 10 men and 10 women if (a) (5 points) The committee must have at least three women. Solution: Break into cases depending on exactly how many women there are: parenleftBigg 10 3 parenrightBiggparenleftBigg 10 2 parenrightBigg + parenleftBigg 10 4 parenrightBiggparenleftBigg 10 1 parenrightBigg + parenleftBigg 10 5 parenrightBiggparenleftBigg 10 0 parenrightBigg (b) (5 points) The committee must have at least three women, but John and Mary cannot both be choosen. Solution: The same three cases. But now subtract the selections that have both John and Mary: bracketleftBiggparenleftBigg 10 3 parenrightBiggparenleftBigg 10 2 parenrightBigg - parenleftBigg 9 2 parenrightBiggparenleftBigg 9 1 parenrightBiggbracketrightBigg + bracketleftBiggparenleftBigg 10 4 parenrightBiggparenleftBigg 10 1 parenrightBigg - parenleftBigg 9 3 parenrightBiggparenleftBigg 9 0 parenrightBiggbracketrightBigg + bracketleftBiggparenleftBigg 10 5 parenrightBiggparenleftBigg 10 0 parenrightBigg - 0 bracketrightBigg Page 1 of 8 Please go on to the next page. . .
2. (10 points) How many arrangements of the letters in STATISTICALLY have all of the fol- lowing properties: (i) at least two consonants between successive vowels (ii) the arrangement starts with a vowel (iii) the consonants are in alphabetical order.

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