Four cards labeled A, B, C, and D are randomly placed in four boxes labeled A, B, C, and D.
Count the number of elements in the event that at least one card is placed in the box with the corresponding letter.
show your work
To count the number of elements in the event that at least one card is placed in the box with the corresponding letter, we need to calculate the number of ways that each card can be placed in its corresponding box, and subtract that from the total number of possible arrangements.
Total number of possible arrangements = 4! = 24
Number of ways at least one card is placed in the correct box:
- Card A in box A: 3 ways (B, C, D in remaining boxes)
- Card B in box B: 3 ways (A, C, D in remaining boxes)
- Card C in box C: 3 ways (A, B, D in remaining boxes)
- Card D in box D: 3 ways (A, B, C in remaining boxes)
Total number of ways at least one card is placed in the correct box = 3 + 3 + 3 + 3 = 12
Therefore, the number of elements in the event that at least one card is placed in the box with the corresponding letter is 12.
no
We can approach this problem by calculating the number of ways that all cards are not placed in their corresponding box and subtracting that from the total number of possible arrangements.
Number of ways all cards are not placed in their corresponding box:
- Card A in box B, C, or D: 3 ways
- Card B in box A, C, or D: 3 ways
- Card C in box A, B, or D: 3 ways
- Card D in box A, B, or C: 3 ways
Number of total ways all cards are not placed in their corresponding box = 3 * 3 * 3 * 3 = 81
Total number of possible arrangements = 4! = 24
Number of elements in the event that at least one card is placed in the box with the corresponding letter = Total possible arrangements - Ways all cards are not placed correctly
= 24 - 81
= 24 - 81
= 24 - 81
= 3
Therefore, there are 3 elements in the event that at least one card is placed in the box with the corresponding letter.