How many different combinations of four letters beginning with E can be formed from the letters of the word equation ?

Options

A. 630
B. 360
C. 210
D. 336

Answer

To solve the problem of how many different combinations of four letters beginning with E can be formed from the letters of the word "equation," we can follow these steps:

### Step 1: Identify the letters in the word "equation"
The letters in the word "equation" are: E, Q, U, A, T, I, O, N. There are a total of 8 letters.

### Step 2: Fix the first letter as E
Since we need to form combinations that begin with the letter E, we will fix E as the first letter. This leaves us with the remaining letters: Q, U, A, T, I, O, N.

### Step 3: Count the remaining letters
After fixing E, we have 7 letters left (Q, U, A, T, I, O, N) to choose from for the remaining 3 positions in our combination.

### Step 4: Choose 3 letters from the remaining 7
We need to select 3 letters from the 7 remaining letters. The number of ways to choose 3 letters from 7 can be calculated using the combination formula:

\[ \text{Number of combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Where:
- \( n \) is the total number of items to choose from (7 in this case)
- \( r \) is the number of items to choose (3 in this case)

So we calculate:

\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \cdot 4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

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

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

### Step 6: Calculate the total combinations
Now, we multiply the number of ways to choose the letters by the number of arrangements:

\[ \text{Total combinations} = \text{Number of ways to choose} \times \text{Number of arrangements} = 35 \times 6 = 210 \]

### Final Answer
Thus, the total number of different combinations of four letters beginning with E that can be formed from the letters of the word "equation" is **210**.

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