Question

There are nine cards each of different colors which are to arranged in a line.These 9 cards include a pink card and a green card.Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.How many different arrangement of 3 cards do not have the pink card next to the green card?

Explain when to use permutation and when to use combination?

Answer 1

consider the pink and green cards stuck together and treated as one item
then we only have to arrange 2 of 8 cards
number of arrangements = 8*7 = 56
But in each of those 56, the pink and blue cards could be reversed, that is , pink, blue OR blue,pink

so the number of case with the cards side by side = 112

total number of cases without restrictions = 9*8*7 = 504

So the number of arrangements they are apart = 504-112 = 392

in simplest terms, for a permutation you consider the order of the items
in a combination, the order in which the event are happening does not matter
e.g. when choosing 3 out of 10 members to form a committee it does not matter
in which order you choose them, <------ combination
but,
if we find the number of 1st, 2nd, and 3rd finishing positions in a 500 m race, the order
does matter <----- arrangement

Answer 2

The answer is::504-28=476.Why are we doing 8*7 and 9*8*7?