To determine the probability that the letters in the word "ENVELOPE" will be rearranged to spell "ENVELOPE" exactly, let's break down the problem step-by-step:
1. Determine the total number of letters in "ENVELOPE":
There are 8 letters in total.
2. Identify the number of occurrences of each letter:
- E: appears 3 times
- N: appears 1 time
- V: appears 1 time
- L: appears 1 time
- O: appears 1 time
- P: appears 1 time
3. Calculate the total number of distinct permutations of the letters in "ENVELOPE":
The formula to find the number of permutations of a multiset (where some elements may be repeated) is given by:
[tex]\[
\text{{Total permutations}} = \frac{{n!}}{{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}}
\][/tex]
where [tex]\(n\)[/tex] is the total number of items to arrange, and [tex]\(n_1, n_2, \ldots, n_k\)[/tex] are the frequencies of the distinct items.
Plugging in the values, we get:
[tex]\[
\text{{Total permutations}} = \frac{{8!}}{{3! \cdot 1! \cdot 1! \cdot 1! \cdot 1! \cdot 1!}}
\][/tex]
4. Calculate the factorial values:
- [tex]\(8! = 40320\)[/tex]
- [tex]\(3! = 6\)[/tex]
- [tex]\(1! = 1\)[/tex] (for N, V, L, O, P)
5. Substitute these values into the formula:
[tex]\[
\text{{Total permutations}} = \frac{{40320}}{{6 \cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1}} = \frac{{40320}}{{6}} = 6720
\][/tex]
6. Determine the number of favorable outcomes:
There is only 1 favorable outcome, which is the exact sequence "ENVELOPE".
7. Calculate the probability:
[tex]\[
\text{{Probability}} = \frac{{\text{{Number of favorable outcomes}}}}{{\text{{Total number of permutations}}}}
\][/tex]
Substituting the values:
[tex]\[
\text{{Probability}} = \frac{{1}}{{6720}}
\][/tex]
8. Choose the correct probability from the given options:
The fraction [tex]\(\frac{1}{6720}\)[/tex] closely matches option (C) which is [tex]\(\frac{1}{2240}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(\frac{1}{2240}\)[/tex]
