Hello everyone. I'm revewing these 2 problems and im' not seeing how they are getting the answer for the "...exactly 4 penny's" Here's the 2 problems: (a) A large pile of coins consist of quarters, dimes, nickels, and pennies (at least 20 of each). How many different collections of 20 coins can be chosen if AT LEAST 4 pennies must be chosen? Well you have 4 categories, Quaters, Dimes, nickles and Pennies it says (at least 20 of each). I'm going to represent the categories as bars |'s and the coins as x's. If the problem said, if at least 4 pennies are already chosen you can assume you already put 4 x's in the penny category, now you have 16 places left to put coins, anywhere you want. so you could have like this: xxxxx|xxxxx|xxxx|xx ------------------- xxxx P N Q D The above ---- means the coins u distrubted, and the stuff under the --- means the coins already there. So if you have 4 categories thats represented by n-1 bars (3) and 16 x's. So you have a total of (16+3 choose 16) = 969 which is the correct answer. Now for the second question: (b) How many different collections of 20 coins can be chosen if EXACTLY 4 pennies must be chosen? THe work shown is, they only used 2 bars |'s and 16 x's and got: (16 + 2 choose 16) = (18 choose 16) = 153. Now why did they only have 3 categories now isntead of 4? The reason I say they only had 3 is because they had 2 bars, thus they had to have 2+1 categories. Why in the first problem did they have to use 4 categories, is it because they said "at least 20 of each?" and in the 2nd problem they only said, If you have exactly 4 penny's, how many different collections of 20 coins can be chosen? I thought to myself, well if only 4 penny's are allowed the minimum amount of categories to use and still put 20 coins in would be to have a category of Penny's which is going to have 4 x's, now your going to have 16 x's (coins) to put in any category you want. So why couldn't u put 4 x's in penny's and 16 in dimes? Then you would only have 2 categories, so 2-1 = 1 bar. Then you would have (16 + 1 choose 16) = (17 choose 16) But the answer is (16+2 choose 16) = (18 choose 16), why did they have to use 3 categories? THanks!