A lady has 4 dresses and she want to leave 2 of them behind and bring the rest with her on the trip. How many ways are there for the lady bring dresses with her?
The Correct Answer and Explanation is:
The problem involves choosing 2 dresses to leave behind out of 4 dresses, meaning the lady will bring the remaining 2 dresses with her. This is a classic example of a combination problem because the order in which the dresses are chosen does not matter, only which dresses are selected.
Step 1: Understand the Problem
The lady has 4 dresses, and she wants to bring 2 of them on the trip. To do this, she will leave 2 dresses behind. The task is to determine in how many ways she can choose 2 dresses to leave behind (which is equivalent to choosing 2 dresses to bring with her). This is a selection problem where the order does not matter.
Step 2: Applying the Combination Formula
The number of ways to choose ( r ) objects from ( n ) objects without regard to the order is given by the combination formula:
[
C(n, r) = \frac{n!}{r!(n – r)!}
]
Where:
- ( n ) is the total number of objects (in this case, 4 dresses),
- ( r ) is the number of objects to choose (in this case, 2 dresses to leave behind).
So, the number of ways the lady can choose which 2 dresses to leave behind is:
[
C(4, 2) = \frac{4!}{2!(4 – 2)!} = \frac{4!}{2!2!} = \frac{4 \times 3 \times 2!}{2! \times 2!} = \frac{12}{4} = 6
]
Step 3: Interpretation of the Result
Thus, the number of ways the lady can leave 2 dresses behind and bring the other 2 dresses with her is 6. This means there are 6 different combinations of 2 dresses she could leave behind.
Step 4: Verifying by Listing Combinations
To further understand, we can list all the possible ways she can leave 2 dresses behind:
- Leave behind dresses 1 and 2, bring 3 and 4.
- Leave behind dresses 1 and 3, bring 2 and 4.
- Leave behind dresses 1 and 4, bring 2 and 3.
- Leave behind dresses 2 and 3, bring 1 and 4.
- Leave behind dresses 2 and 4, bring 1 and 3.
- Leave behind dresses 3 and 4, bring 1 and 2.
Thus, the total number of ways is indeed 6.
Conclusion
Therefore, there are 6 ways for the lady to choose which 2 dresses to leave behind and bring the other 2 with her on her trip. This is a combination problem where the order of selection does not matter.