Originally posted by Dumbarse
1) How many different six letter words can be formed from the letters of the word AUSTRALIA ??
2) How many different sums of money can be made up from six $20 notes, two $5 notes, four $2 coins, seven $1 coins, three 50c coins, and one 20c coin ??
1) There are 3 'A's in australia, 7 different characters altogether.
So this is like the other problem that fuelled the huge argument in the FREQUENT thread. The solution is to break it up into cases:
No 'A's: 6 different letters, all must be included: 6!
1 'A': 7 different letters, 'A' must be included:
'A' can be in any of the six, then for the remaining 5 slots there are 6, 5, ..., 2 options remaining: so it's 6*6*5*4*3*2
2 'A's: Choose two slots for the two 'A's, then 6, 5, ..., 3 options remaining for the 4 other slots: (6C2)*6*5*4*3
3 'A's: Same idea: (6C3)*6*5*4
Altogether: 12840
2) The strategy here is to determine how many different amounts between 0 to $20, you can make (i.e look at all the possible remainders when you divide by 20). Since this can be repeated when you add a $20 note, for all the way up to $120. Then you must determine how many different options above $120 you can make:
One 20c coin, more than enough 50c coins: can only make $x.00, $x.20, $x.50, $x.70 (4 choices)
More than enough $1 and $2 coins: unit figure can be anything from 0 to 9 (10 choices)
2 $5 notes: can make anything from 0 to 19 (mod 20) (multiply choice by 2)
So there are 4 * 10 * 2 possible remainders, and since you can have 0, 1, 2, ..., 5, you multiply by 6: total 480.
Now, you've got to determine how many different options above $120 you can make.
Now whether you have 5 or 6 $20 notes, it won't give you any more options, so lets assume 6
So must determine how many options you can make WITHOUT the twenties notes:
From less than $20: 80 options (determined above)
More than $20: can make up to $25 using gold coins. (6*4 = 24)
$26 or over: can make $26.00, $26.20, $26.50, $26.70 (=4)
So 108 options above $120
Total options (including zero sum as an option) is 588
Excluding zero as an option: 587
Edit: which is the same as blackjack, which is cool
