How many words can be formed from the letters of the word 'ENGINEER' ?

Question

Accepted Answer

To determine how many words can be formed from the letters of the word "ENGINEER," we need to consider the arrangement of its letters, taking into account any repetitions.

The word "ENGINEER" consists of 8 letters, where:
- E appears 3 times
- N appears 2 times
- G, I, and R each appear 1 time

The formula for calculating the number of distinct permutations of a word with repeated letters is given by:

$$ \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} $$

where $ n $ is the total number of letters, and $ p_1, p_2, \ldots, p_k $ are the frequencies of the repeated letters.

For "ENGINEER":
- $ n = 8 $ (total letters)
- The frequency of E is 3, so $ p_1 = 3 $
- The frequency of N is 2, so $ p_2 = 2 $

Plugging these values into the formula, we get:

$$ \frac{8!}{3! \times 2!} $$

Calculating each component:
- $ 8! = 40320 $
- $ 3! = 6 $
- $ 2! = 2 $

Now, substitute these into the formula:

$$ \frac{40320}{6 \times 2} = \frac{40320}{12} = 3360 $$

Therefore, the number of distinct words that can be formed from the letters of "ENGINEER" is 3360.

### Explanation of Options:

1. **Option 1 (3360):** This is the correct answer as calculated above using the permutation formula for words with repeated letters.

2. **Option 2 (40320):** This would be the total number of permutations if all letters were unique. Since there are repeated letters, this option does not account for the repetitions and is therefore incorrect.

3. **Option 3 (20160):** This option is incorrect as it does not match the calculated value of 3360. It seems to be a miscalculation or a misunderstanding of the formula application.

4. **Option 4 (6720):** This option is also incorrect. It might result from a partial calculation or an error in considering the repetitions of letters.

In conclusion, Option 1 (3360) is the correct answer, as it accurately reflects the number of distinct permutations of the letters in "ENGINEER" considering the repetitions.