counting_prob_solns_2010 - counting problems. "let me...

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counting problems. .. "let me count the ways" 1. How many car license plates can a state have if each plate consists of 2 letters followed by 4 digits? (Use the alphabet with 26 letters.) 26 * 26 * 10 4 = 6,760,000 How many car license plates can the state have if each plate consists of 3 letters followed by 3 digits? 26 3 * 10 4 = 17,576,000 How many car license plates can the state have if both patterns are allowed? 6,670,000 + 17,576,000 = 24,246,000 How many car license plates are possible for the current California pattern of 1 digit, 3 letters, 3 numbers ? (You may ignore the existence of personalized plates for this problem) (Don't worry about removing sequences of letters that spell obnoxious words like BAD.) 10 * 26 * 26 * 26 * 10 * 10 * 10 = 175,760,000 2. How many numbers are possible with the current US Social Security Number pattern of 9 digits? 10 9 = 1,000,000,000 1 billion (if we don't wipe out life on the planet first, we might eventually run out!) 3. A club has 10 members, 5 men and 5 women. How many ways can they appoint a committee of 4 members to plan an event? " 10 choose 4" 10! / [ 6! * 4! ] = 10 * 9 * 8 * 7 * 6! / [6! * 4 * 3 * 2 * 1] = 10 * 9 * 8 * 7 / 4 * 3 * 2 3 = 10 * 9 \ * 8 / * 7 / 4 / * 3 \ * 2 / = 10 * 21 = 210
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How many ways can they appoint a committee of 4 members if all the members are chosen from among the women? "5 chose 4" = 5! / [4! * 1! ] = 5 * 4! / [4! * 1] = 5 If the committee is appointed at random, what is the probability that it consists of 4 women? 5 / 210 = .0238 4. A small pizza parlor offers five toppings for its pizzas. (All pizzas include tomato sauce and cheese.) How many different pizzas with three toppings can be made? (No double orders of the same ingredient allowed!) "5 chose 3" = 5! / [3! * 2!] = 5 * 4 * 3! / [3! * 2] = 5 * 4 / 2 = 10 How many different pizzas in all can be made, again if no double orders of the same ingredient are allowed? (Try to find a way to calculate this all at once rather than considering separate cases for no toppings, 1 topping, 2 toppings, etc. You may think of the ingredients lined up in a row and picking or not picking each one) this can be worked out by applying the general multiplication rule topping 1 Yes or No * topping 2 Yes or No and so on 2 5 = 32
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more counting problems and some related questions these are less important than the other skills, but for those of you who've asked for some more practice, here are some to try. some of them are tricky, so don't be concerned if you can't do them all
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This note was uploaded on 07/29/2010 for the course PH 141 taught by Professor Lahiff during the Summer '10 term at University of California, Berkeley.

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counting_prob_solns_2010 - counting problems. "let me...

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