Words are formed with the letters of the word PEACE. Find the probability that 2 E's come together.

Answer

To solve the problem of finding the probability that the two E's in the word "PEACE" come together, we will follow these steps:

### Step 1: Determine the total arrangements of the letters in "PEACE".
The word "PEACE" consists of 5 letters where the letter E appears twice. To find the total arrangements of these letters, we use the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{p_1! \cdot p_2! \cdots p_k!} \]

Where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

For "PEACE":
- Total letters \( n = 5 \) (P, E, A, C, E)
- The letter E appears 2 times.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

### Step 2: Determine the favorable arrangements where the two E's are together.
To find the arrangements where the two E's are together, we can treat the two E's as a single entity or block. This gives us the following letters to arrange: (EE), P, A, C.

Now we have 4 entities to arrange: (EE), P, A, C.

The number of arrangements of these 4 entities is:

\[ \text{Favorable arrangements} = 4! = 24 \]

### Step 3: Calculate the probability.
The probability that the two E's come together is given by the ratio of the favorable arrangements to the total arrangements:

\[ \text{Probability} = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{24}{60} = \frac{2}{5} \]

Thus, the final answer is:

\[ \text{Probability} = \frac{2}{5} \]

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