All possible 120 permutations of WDSMC are arranged in dictionary order, as if each were an ordinary five- letter word. The last letter of the ` 86^(th)` word in the list, is :

Options

A. W
B. D
C. M
D. C

Answer

To find the last letter of the 86th permutation of the letters in "WDSMC" when arranged in dictionary order, we will follow these steps:

### Step 1: Arrange the letters in alphabetical order
The letters in "WDSMC" are C, D, M, S, and W. Arranging them in alphabetical order gives us:
- C, D, M, S, W

### Step 2: Calculate the number of permutations starting with each letter
We will calculate how many permutations start with each letter until we reach or exceed the 86th permutation.

1. **Starting with C**:
- Remaining letters: D, M, S, W
- Number of permutations = 4! = 24

2. **Starting with D**:
- Remaining letters: C, M, S, W
- Number of permutations = 4! = 24
- Total permutations so far = 24 (C) + 24 (D) = 48

3. **Starting with M**:
- Remaining letters: C, D, S, W
- Number of permutations = 4! = 24
- Total permutations so far = 48 (C and D) + 24 (M) = 72

4. **Starting with S**:
- Remaining letters: C, D, M, W
- Number of permutations = 4! = 24
- Total permutations so far = 72 (C, D, M) + 24 (S) = 96

### Step 3: Determine which letter starts the 86th permutation
Since the total permutations starting with S exceed 86, we will focus on permutations starting with S.

### Step 4: Calculate permutations starting with S
Now we need to find the permutations starting with S and then the next letter.

1. **Starting with SC**:
- Remaining letters: D, M, W
- Number of permutations = 3! = 6
- Total permutations so far = 72 + 6 = 78

2. **Starting with SD**:
- Remaining letters: C, M, W
- Number of permutations = 3! = 6
- Total permutations so far = 78 + 6 = 84

3. **Starting with SM**:
- Remaining letters: C, D, W
- Number of permutations = 3! = 6
- Total permutations so far = 84 + 6 = 90

Since 86 falls between 84 and 90, we know that the 86th permutation starts with SM.

### Step 5: Determine the next letters after SM
Now we need to find the permutations starting with SM.

1. **Starting with SMC**:
- Remaining letters: D, W
- Number of permutations = 2! = 2
- Total permutations so far = 84 + 2 = 86

The 86th permutation is SMC followed by the remaining letters in alphabetical order, which are D and W.

### Step 6: Identify the 86th permutation
The 86th permutation is:
- SMCW

### Step 7: Find the last letter of the 86th permutation
The last letter of the 86th permutation "SMCW" is:
- **D**

### Summary
The last letter of the 86th word in the list is **D**.

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