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In how many ways can the numbers 1 through 5 be entered once each into the five boxes below so that all the given inequalities are true?

 Mar 4, 2023
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To satisfy the given inequalities, we need to place the numbers 1 through 5 into the boxes such that:

- Box 1 < Box 2
- Box 2 > Box 3
- Box 3 < Box 4
- Box 4 > Box 5

We can approach this problem by considering each box in turn and counting the number of possible values that can be placed in that box.

Box 1: There are five possible values that can be placed in the first box.

Box 2: Since the number in Box 2 must be greater than the number in Box 1, there are only four possible values that can be placed in the second box.

Box 3: Since the number in Box 3 must be less than the number in Box 2, there are only three possible values that can be placed in the third box.

Box 4: Since the number in Box 4 must be greater than the number in Box 3, there are only two possible values that can be placed in the fourth box.

Box 5: Since the number in Box 5 must be less than the number in Box 4, there is only one possible value that can be placed in the fifth box.

Therefore, the total number of ways to enter the numbers 1 through 5 into the five boxes is:

5 x 4 x 3 x 2 x 1 = 120

Therefore, there are 120 different ways to enter the numbers 1 through 5 into the five boxes so that all the given inequalities are true.

 Mar 5, 2023

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