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?
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
The answer is::504-28=476.Why are we doing 8*7 and 9*8*7?
To find the number of different arrangements of 3 cards that do not have the pink card next to the green card, we need to subtract the number of arrangements that have the pink card next to the green card from the total number of arrangements.
First, let's calculate the total number of arrangements of 3 cards from the 9 cards. We can use permutations because the order of the cards matters.
In this case, we have 9 cards to choose from and we are selecting 3 cards. So, we use the formula for permutations: nPr = n! / (n - r)!
Here, n = 9 (the total number of cards) and r = 3 (the number of cards we are selecting).
Total number of arrangements = 9P3 = 9! / (9 - 3)! = 9! / 6! = 9 × 8 × 7 = 504.
Now, let's calculate the number of arrangements that have the pink card next to the green card.
To have the pink card next to the green card, we can treat the pink and green cards as a single entity. So, now we have 8 entities to arrange (7 single cards and the pink-green combination).
Using the same permutation formula, we find the number of arrangements: 8P3 = 8! / (8 - 3)! = 8! / 5! = 8 × 7 × 6 = 336.
Finally, to find the number of arrangements that do not have the pink card next to the green card, we subtract the number of arrangements with the pink card next to the green card from the total number of arrangements:
Number of arrangements without the pink card next to the green card = Total number of arrangements - Number of arrangements with the pink card next to the green card
= 504 - 336
= 168.
Therefore, there are 168 different arrangements of 3 cards from the 9 cards that do not have the pink card next to the green card.
Now, let's discuss when to use permutation and when to use combination.
Permutations are used when the order of the elements matters, and combination is used when the order does not matter.
In this problem, we need to arrange the cards in a line, and the order matters, so we use permutations. However, if the question asked for the number of different groups of 3 cards we can select without considering the order, we would use combinations.
To determine when to use permutation or combination, we need to understand the difference between the two concepts.
Permutations are used when the order or arrangement of the elements matters. It calculates all possible arrangements of selecting items from a set. The formula for permutations is given by:
nPr = n! / (n-r)!
where n is the total number of items and r is the number of items being selected.
Combinations, on the other hand, are used when the order doesn't matter. It calculates the total number of ways to select items from a set without considering their arrangement. The formula for combinations is given by:
nCr = n! / (r! * (n-r)!)
where n is the total number of items and r is the number of items being selected.
Now, let's apply these concepts to the given question.
We have 9 cards, and we need to select 3 cards from them and arrange them in a line. We want to find the number of different arrangements that do not have the pink card next to the green card.
To solve this problem, we will use the principle of permutations and combinations together.
First, let's select the position for the pink card. We have 9 positions to choose from, so we have 9 choices.
Next, we need to choose the position for the green card. Since we don't want the pink and green cards to be next to each other, we have 8 remaining positions to choose from.
After placing the pink and green cards, we have 7 remaining cards to choose from for the third position in the line.
Using the permutation formula, we can calculate the number of ways to arrange the remaining 6 cards in the remaining 6 positions:
6P6 = 6! / (6-6)! = 6! / 0! = 6!
Finally, to find the total number of different arrangements that do not have the pink card next to the green card, we multiply the choices we made for each step:
Number of arrangements = 9 * 8 * 6! = 9 * 8 * 6!
Therefore, the total number of different arrangements of 3 cards that do not have the pink card next to the green card is 9 * 8 * 6!.