How many words can be formed from the all letters of word "ASSESSMENT" ?

Question

Accepted Answer

To determine how many words can be formed from the letters of the word "ASSESSMENT," we need to consider the total number of letters and the repetitions of specific letters.

The word "ASSESSMENT" consists of 10 letters. The breakdown of the letters is as follows:
- A: 1
- S: 4
- E: 2
- M: 1
- N: 1
- T: 1

The formula for calculating the number of permutations of a multiset (a set with repeated elements) 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.

Applying this formula to "ASSESSMENT":

$$ n = 10 $$ (total letters)

The frequencies of the repeated letters are:
- S: 4
- E: 2

Thus, the calculation is:

$$ \frac{10!}{4! \times 2!} $$

Calculating each component:

- $ 10! = 3,628,800 $
- $ 4! = 24 $
- $ 2! = 2 $

Now, substitute these values into the formula:

$$ \frac{3,628,800}{24 \times 2} = \frac{3,628,800}{48} = 75,600 $$

Therefore, the number of distinct words that can be formed from the letters of "ASSESSMENT" is 75,600.

**Explanation of Options:**

1. **Option 1: 21,600** - This number is incorrect because it underestimates the total permutations by not properly accounting for the factorial division of repeated letters.

2. **Option 2: 30,240** - This number is incorrect as it does not match the calculated permutations when considering the repetitions of 'S' and 'E'.

3. **Option 3: 151,200** - This number is incorrect because it overestimates the permutations, likely by not fully accounting for the repeated letters.

4. **Option 4: 75,600** - This is the correct answer, as shown by the detailed calculation above.

In conclusion, the correct number of distinct words that can be formed from the letters of "ASSESSMENT" is 75,600, which corresponds to Option 4.