combinations/permutations/algorithm/formula? please?

Question

i know the factorial part of this answer (84)... i need specific positions of 6 people (A-F) in nine chairs (1-9)... there are 84 seating combinations possible... i need to know the 84 different seating arrangements for 6 people (A-F) sitting at a table with nine chairs (1-9)... i think i used to know how to write such a proggie in BASIC but that was long years ago... i tried to do it by writing the combinations on paper but i could never get 84...

2:18pm... Brambly... i said i knew the factorial part of the answer... i'd love it if you would go on with the list you started and show me IF you get 84 combinations... thanks bud...

Answer 1

The answer is 9! / (6!*3!).

You're dealing with combinations and not permutations, so we don't care which person is which... you really have nine chairs, six of which are occupied and three of which are empty. Forget about ABCDEF being the six people: Let X represent an occupied chair, and O an empty chair.

If you need to create a list of all possible combinations, just start with the X'es all to the left and the O's to the right, then do a bubble sort thingy.

XXXXXXOOO
XXXXXOXOO
XXXXXOOXO
XXXXXOOOX
XXXXOXXOO
XXXXOXOXO
XXXXOXOOX
XXXXOOXXO
XXXXOOXOX
XXXXOOOXX
etc.

If you grind your way through the whole list, you'll find there are 84 combinations. There's some neat stuff going on here with triangular numbers: you can see that there's one arrangement where the first O is in the 7th position, three where the first O is in the 6th position, six where the first O is in the 5th position... follow the pattern and you'll see that there are 1+3+6+10+15+21+28 combinations in the list. That adds up to 84.

Hopefully that helps!

Answer 2

TI 83 read the manuel