You own 5 pairs of jeans and want to take 2 of them with you on vacation. In how many ways can you choose 2 pairs of jeans?
There are two ways to approach this problem: using the permutation formula or using combinatory reasoning.
Permutation formula:
The number of permutations of n distinct objects taken k at a time is given by:
nP(k) = n! / (n - k)!
where n! (read as "n factorial") is the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
In this case, we want to take 2 pairs of jeans out of 5. Therefore, n = 5 and k = 2. Plugging these values into the formula, we get:
nP(k) = 5P2 = 5! / (5 - 2)! = 5 x 4 x 3 / 3 x 2 x 1 = 10
Therefore, there are 10 ways to choose 2 pairs of jeans out of 5.
Combinatory reasoning:
Another way to solve this problem is to use combinatory reasoning. We can think of it as follows: we have 5 jeans, and we want to choose 2 of them. This means that we need to form pairs of jeans. The first pair can be chosen from any of the 5 jeans, and the second pair can be chosen from the remaining 4 jeans (since we cannot choose the same pair twice). Therefore, the total number of ways to choose 2 pairs of jeans out of 5 is:
5 x 4 / 2 x 1 = 10
where we divide by 2 x 1 to account for the fact that the order of the pairs does not matter.
Therefore, both methods give the same result: there are 10 ways to choose 2 pairs of jeans out of 5.
To calculate the number of ways you can choose 2 pairs of jeans out of 5, you can use the combination formula:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of jeans you own and r is the number of jeans you want to choose.
In this case, n = 5 (total number of jeans) and r = 2 (number of jeans to choose).
Plugging in the values, the formula becomes:
C(5, 2) = 5! / (2! * (5 - 2)!)
C(5, 2) = 5! / (2! * 3!)
Now, let's calculate the factorials:
5! = 5 * 4 * 3 * 2 * 1 = 120
2! = 2 * 1 = 2
3! = 3 * 2 * 1 = 6
Now substitute the factorials back into the formula:
C(5, 2) = 120 / (2 * 6)
C(5, 2) = 120 / 12
C(5, 2) = 10
Therefore, you can choose 2 pairs of jeans from your 5 pairs in 10 different ways.
To find the number of ways you can choose 2 pairs of jeans out of the 5 pairs that you own, you can use the formula for combinations.
The formula for combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
where n is the total number of items in the set, and r is the number of items to be chosen.
In this case, you have 5 pairs of jeans, which means you have a total of 5 items in the set. You want to choose 2 pairs of jeans, so r = 2.
Plugging these values into the formula:
C(5, 2) = 5! / (2! * (5 - 2)!)
Now, let's calculate the values of the factorials:
5! = 5 x 4 x 3 x 2 x 1 = 120
2! = 2 x 1 = 2
(5 - 2)! = 3! = 3 x 2 x 1 = 6
Now, substitute these values into the formula:
C(5, 2) = 120 / (2 x 6)
Simplifying further:
C(5, 2) = 120 / 12
C(5, 2) = 10
Therefore, there are 10 ways you can choose 2 pairs of jeans out of the 5 pairs that you own.