Question

How many linear orders are there on a deck of cards with 2 red, 4 blue, 10 white, and 5 green cards, which do not start and end with the same card? Two orders are considered the same if they have the same color pattern.

Answer 1

number of linear order with 2 red, 4 blue, 10 white, and 5 green cards =21!/(2!*4!*10!*5!)=2444321880

number of linear order with same card on both end=N(red on both end)+N(blue on both end)+N(white on both end)+N(green on both end)

=19!/(0!*4!*10!*5!)+19!/(2!*2!*10!*5!)+19!/(2!*4!*8!*5!)+19!/(2!*4!*10!*3!)

=11639628+69837768+523783260+116396280 =721656936

therefore linear orders which do not start and end with the same card =2444321880-721656936

=17722664944