The letters o f the word FATHER are written on separate cards, two cards are drawn at random. Probability that both are vowels is

Options

A. `(2)/(15)`
B. `(1)/(15)`
C. `(3)/(15)`
D. `(1)/(15)`

Answer

To solve the problem of finding the probability that both cards drawn from the letters of the word "FATHER" are vowels, we will follow these steps:

### Step 1: Identify the total number of letters
The word "FATHER" consists of the letters F, A, T, H, E, R. Therefore, the total number of letters is 6.

### Step 2: Determine the total number of ways to draw 2 cards
To find the total number of ways to choose 2 cards from 6, we use the combination formula:
\[ \text{Total ways} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

### Step 3: Identify the vowels in the word
The vowels present in the word "FATHER" are A and E. Therefore, there are 2 vowels in total.

### Step 4: Determine the number of favorable outcomes
We need to find the number of ways to choose 2 vowels from the 2 available vowels (A and E). This can be calculated as:
\[ \text{Favorable ways} = \binom{2}{2} = 1 \]

### Step 5: Calculate the probability
The probability of drawing 2 vowels is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ \text{Probability} = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{1}{15} \]

### Final Answer
Thus, the probability that both cards drawn are vowels is \(\frac{1}{15}\).

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