In how many ways can the 7 letters A, B, C, D, E, F, and G be arranged so that C and E are never together?

Answer

To find the number of ways to arrange the letters A, B, C, D, E, F, and G such that C and E are never together, we can follow these steps:

### Step 1: Calculate the total arrangements of the 7 letters
The total number of ways to arrange 7 distinct letters is given by the factorial of the number of letters.

\[ \text{Total arrangements} = 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

### Step 2: Calculate the arrangements where C and E are together
To find the arrangements where C and E are together, we can treat C and E as a single unit or block. Therefore, we will have the following blocks to arrange: (CE), A, B, D, F, G. This gives us a total of 6 blocks to arrange.

\[ \text{Arrangements with C and E together} = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

### Step 3: Account for the arrangements of C and E within their block
Since C and E can be arranged within their block in 2 different ways (CE or EC), we need to multiply the arrangements of the blocks by 2.

\[ \text{Total arrangements with C and E together} = 6! \times 2 = 720 \times 2 = 1440 \]

### Step 4: Calculate the arrangements where C and E are never together
To find the arrangements where C and E are never together, we subtract the arrangements where C and E are together from the total arrangements.

\[ \text{Arrangements where C and E are never together} = 7! - \text{(Arrangements with C and E together)} \]

\[ = 5040 - 1440 = 3600 \]

### Final Answer
Thus, the number of ways to arrange the letters A, B, C, D, E, F, and G such that C and E are never together is **3600**.

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