Nine cards, each of a different colour, are to be arranged in a line.

The 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.

Q.How many of the arrangements of 3 cards contain the pink card?

To find the number of arrangements of 3 cards that contain the pink card, we can break it down into two cases: when the pink card is in the first position and when the pink card is in one of the other two positions.

Case 1: The pink card is in the first position.
In this case, we need to choose 2 cards from the remaining 8 cards (excluding the pink card). The number of ways to do this is given by the combination formula C(n, k), which represents the number of ways to choose k elements from a set of n elements. In this case, n = 8 (remaining cards) and k = 2 (cards to be chosen). So, C(8, 2) = 8! / (2! * (8-2)!) = 8! / (2! * 6!) = (8 * 7) / (2 * 1) = 28.

Case 2: The pink card is in one of the other two positions.
In this case, we again need to choose 2 cards from the remaining 8 cards (excluding the pink card). The number of ways to do this is also C(8, 2) = 28.

To find the total number of arrangements that contain the pink card, we sum up the arrangements from both cases:
Total = Case 1 + Case 2 = 28 + 28 = 56.

Therefore, there are 56 arrangements of 3 cards that contain the pink card.