Difference between at least and exactly? combinatorics

[mr_coffee]

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 that's 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!

 

[mattmns]

The reason is this. In the first problem you choose 4 pennies, then you have 16 coins left to pick, and these coins can be any of the 4 choices (quarters, dimes, nickels, pennies). In the second problem you choose 4 pennies, then you have 16 coins left to pick, but none of these remaining 16 coins can be pennies, thus you are picking from 3 (quarters, dimes, nickels).

 

[mr_coffee]

Oooo hah i got it!
thanks again matt!