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HTUT-382 - We have a set of 5 numbers: 1, 2, 3, 4, and 5. These numbers can be placed in an order that either includes or excludes the number 5. Since including or excluding the number 5 affects the total number of possible arrangements, we need to calculate the number of possible arrangements for both scenarios and then sum them up to get the total number of possible arrangements.### **Case 1: Including the number 5**When we include the number 5, we have to place all 5 numbers in the arrangement. This means we need to determine the number of possible arrangements (permutations) of 5 distinct items. The formula for permutations of N items is: **`P(N) = N!`**Plugging in the number of items:**`P(5) = 5! = 5 × 4 × 3 × 2 × 1 = 120`**So, there are **120** possible arrangements when including the number 5.### **Case 2: Excluding the number 5**When we exclude the number 5, we have only 4 numbers to arrange: 1, 2, 3, and 4. This means we need to determine the number of possible arrangements (permutations) of 4 distinct items. The formula for permutations of N items is:**`P(N) = N!`**Plugging in the number of items:**`P(4) = 4! = 4 × 3 × 2 × 1 = 24`**So, there are **24** possible arrangements when excluding the number 5.### **Total Number of Possible Arrangements**To get the total number of possible arrangements, we need to sum the number of arrangements from both cases (including AND excluding the number 5):**`Total arrangements = Number of arrangements including 5 + Number of arrangements excluding 5`****`Total arrangements = 120 + 24 = 144`**Therefore, the total number of possible arrangements is **144**.**Final Answer: 144**
2020年2月14日
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