Coins Two
How many ways are there of distributing 9 different coins into two pocket?.?
Well it depends.
1.) If it matters which coin is in the left pocket and which in the right (which is probably not the case), then
you can either have 9 coins in the left pocket and none in the right one, 8 coins in the left pocket and 1 coin in the right... etc
When we make our choice for the left pocket, the coins in the right pocket are determined as well. So we can just say that the number of ways are
9 coins in the left pocket + 8 coins in the left pocket + 7 coins ... + 1 coin in the left pocket + 0 coins in the left pocket
Because the order of the coins inside of the pocket doesn't matter, we use the formula for "choosing n coins out of 9 coins". And the number of ways are:
(9 \choose 9) + (9 \choose 
+ (9 \choose 7) + (9 \choose 6) + (9 \choose 5) + (9 \choose 4) + (9 \choose 3) + (9 \choose 2) + (9 \choose 1) + (9 \choose 0) =
9!/9!0! + 9!/8!1! + 9!/7!2! + 9!/6!3! + 9!/5!4! + 9!/4!5! + 9!/3!6! + 9!/2!7 + 9!/1!8! + 9!/0!9! =
1 + 9 + 36 + 84 + 126 + 126 + 84 + 36 + 9 + 1 =
512 ways
+++++++++
Note that these are binomial coefficients and you can quickly obtain this sum as (1+1)^9 = 2^9 = 512
+++++++++
2.) If we don't think it is important which pocket is which, then the number of ways is
512/2 = 246 ways
(because then 5 coins in the left pocket + 4 in the right is exactly the same as 4 in the left and 5 in the right, so we have counted everything twice and now have to divide the result by 2)
[I am calling them the left and the right pocket... could also name them the first and the second pocket...]
Hope this helps.
P.S.: I see some people are giving answers of the sort 9·9·9·9... this is plain ridiculous. Once you have chosen one of the 9 coins to put in either of the pockets, you only have 8 coins left etc...
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