All letters of the word 'CEASE' are arranged randomly in a row, then the probability that 2 E are found together is

Options

A. ` (7 ) /( 5 ) `
B. ` ( 3 ) / ( 5 ) `
C. ` (2 ) /( 5 ) `
D. ` ( 1 )/ ( 5 ) `

Explanation

To find the probability that the two E's in the word "CEASE" are together, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in "CEASE"
The word "CEASE" consists of 5 letters where 'E' appears twice. The formula for the total arrangements of letters when there are repetitions is given by:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

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

For "CEASE":
- Total letters \( n = 5 \)
- The letter 'E' appears 2 times.

So, the total arrangements are:

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

### Step 2: Calculate the arrangements where the two E's are together
To treat the two E's as a single unit, we can think of "EE" as one letter. Thus, we now have the letters: "EE", "C", "A", "S".

Now we have 4 units to arrange: "EE", "C", "A", "S".

The arrangements of these 4 units are:

\[ \text{Arrangements with EE together} = 4! = 24 \]

### Step 3: Calculate the probability
The probability that the two E's are together is given by the ratio of the favorable outcomes (where E's are together) to the total outcomes.

\[ \text{Probability} = \frac{\text{Arrangements with EE together}}{\text{Total arrangements}} = \frac{24}{60} = \frac{2}{5} \]

### Final Answer
Thus, the probability that the two E's are found together is:

\[ \frac{2}{5} \]

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