Find the number of four letter arrangements of the letters of the word ' SHOOT' . How many of them begin with O?

Answer

To solve the problem of finding the number of four-letter arrangements of the letters in the word "SHOOT" that begin with the letter 'O', we can follow these steps:

### Step 1: Identify the letters in the word "SHOOT"
The letters in the word "SHOOT" are S, H, O, O, T.

### Step 2: Fix the first letter
Since we want the arrangements to begin with 'O', we fix 'O' as the first letter. This leaves us with the letters S, H, O, and T to arrange in the remaining three positions.

### Step 3: Count the remaining letters
After fixing 'O', we have the remaining letters: S, H, O, T. We have one 'O' left and three other distinct letters (S, H, T).

### Step 4: Choose 3 letters from the remaining letters
We need to select 3 letters from the remaining letters (S, H, O, T). The total number of ways to choose 3 letters from these 4 letters can be calculated using combinations:

\[ \text{Number of ways to choose 3 letters from 4} = \binom{4}{3} = 4 \]

### Step 5: Arrange the chosen letters
Once we have chosen 3 letters, we need to arrange them. The number of arrangements of 3 letters is given by \(3!\):

\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 6: Calculate the total arrangements
Now, to find the total number of arrangements that begin with 'O', we multiply the number of ways to choose the letters by the number of arrangements of those letters:

\[ \text{Total arrangements} = \text{Number of ways to choose letters} \times \text{Number of arrangements} = 4 \times 6 = 24 \]

### Final Answer
Thus, the total number of four-letter arrangements of the letters of the word "SHOOT" that begin with 'O' is **24**.

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